Reconfigurable qubit entanglement system

Abstract

According to some embodiments, a system includes a first input coupled to a first qubit and a first switch, the first switch including a first output, a second output, and a third output. The system further includes a first single qubit measurement device coupled to the first output of the first switch and a second single qubit measurement device coupled to the first output of the second switch. The system further includes a first two-qubit measurement device coupled to the second output of the first switch and the second output of the second switch, and a second two-qubit measurement device coupled to the third output of the first switch and the third output of the second switch.

Description

[Technical Field]

[0001]

[0001] Cross-reference of related applications This application claims priority to U.S. Provisional Application No. 63 / 140,784, filed on 22 January 2021, entitled “Fusion-Based Quantum Computing,” and U.S. Provisional Application No. 63 / 293,592, filed on 23 December 2021, entitled “Reconfigurable Qubit Fusion System,” each of which is incorporated in its entirety by reference.

[0002]

[0002] One or more embodiments of this disclosure generally relate to quantum technology devices (e.g., hybrid electronic / photonic devices), more specifically to quantum technology devices for generating entangled states of qubits (e.g., entangled states that can be used as resources for quantum computing, quantum communication, quantum measurement, and other quantum information processing tasks), and to systems and methods for generating syndrome graph data that can be used for quantum error correction in fault-tolerant quantum computing systems. One or more embodiments of this disclosure generally relate to quantum computing devices and methods, more specifically to fault-tolerant quantum computing devices and methods. [Background technology]

[0003]

[0003] In fault-tolerant quantum computing, quantum error correction is required to avoid the accumulation of qubit errors that would subsequently lead to incorrect computation results. One way to achieve fault tolerance is to use error correction codes (e.g., topological codes) for quantum error correction. More specifically, a set of physical qubits can be generated in an entangled state (also referred to herein as an error correction code) that encodes with respect to a single logical qubit that is protected from error.

[0004]

[0004] In some quantum computing systems, cluster states of multiple qubits, or more generally, graph states, can be used as error correction codes. A graph state is a highly entangled multi-qubit state that can be visually represented as a graph having nodes representing qubits and edges representing entanglement between qubits. However, various problems that hinder the generation of entangled states or break up entanglement once it has been generated have hindered the progress of quantum technology that relies on the use of highly entangled quantum states.

[0005]

[0005] Furthermore, in some qubit architectures, such as photonic architectures, the generation of entangled states of multiple qubits is an inherently probabilistic process that may have a low success rate.

[0006]

[0006] Therefore, improved systems and methods for quantum computing that do not necessarily depend on large cluster states of qubits are still needed. [Overview of the Initiative]

[0007]

[0007] This specification describes embodiments of a reconfigurable qubit entanglement system according to one or more embodiments.

[0008]

[0008] According to some embodiments, the method may include the steps of: receiving a plurality of quantum systems, each of which quantum systems comprises a plurality of entangled quantum subsystems, and each of which quantum systems is an independent quantum system that is not entangled with one another; performing a plurality of destructive batch measurements (such as a fusion operation) on different quantum subsystems from each of which quantum systems, wherein the destructive batch measurements destroy the different quantum subsystems and generate batch measurement result data to transfer quantum state information from the different quantum subsystems to other unmeasured quantum subsystems from the plurality of quantum systems; and determining logical qubit states based on the batch measurement result data. The logical qubit states may be determined in a fault-tolerant manner.

[0009]

[0009] According to some embodiments, the method may include the steps of: receiving a plurality of quantum systems, each of which quantum systems comprises a plurality of entangled quantum subsystems, and each of which quantum systems is an independent quantum system that is not entangled with one another; performing a logical qubit gate by performing a plurality of destructive batch measurements (such as a fusion operation) on different quantum subsystems from each of the plurality of quantum systems, wherein the destructive batch measurements destroy the different quantum subsystems and generate batch measurement result data to transfer quantum state information from the different quantum subsystems to other unmeasured quantum subsystems from the plurality of quantum systems; and determining the result of the logical qubit gate based on the batch measurement result data. The result of the logical qubit gate may be determined in a fault-tolerant manner.

[0010]

[0010] According to some embodiments, a quantum computing device may comprise: a qubit entanglement system that generates a plurality of quantum systems, wherein each quantum system of the plurality of quantum systems includes a plurality of quantum subsystems in an entangled state, and each quantum system of the plurality of quantum systems is an independent quantum system that is not entangled with one another; a qubit fusion system that performs a plurality of destructive batch measurements on different quantum subsystems from each of the plurality of quantum systems, wherein the destructive batch measurements destroy the different quantum subsystems and generate batch measurement result data to transfer quantum state information from the different quantum subsystems to other unmeasured quantum subsystems from the plurality of quantum systems; and a classical computing system for determining logical qubit states based on the batch measurement result data.

[0011]

[0011] According to some embodiments, a quantum computing device may comprise: a qubit entanglement system that generates a plurality of quantum systems, wherein each quantum system of the plurality of quantum systems includes a plurality of quantum subsystems in an entangled state, and each quantum system of the plurality of quantum systems is an independent quantum system that is not entangled with one another; a qubit fusion system that executes a logic qubit gate by performing a plurality of destructive batch measurements on different quantum subsystems from each of the plurality of quantum systems, wherein the destructive batch measurements destroy different quantum subsystems and generate batch measurement result data to transfer quantum state information from different quantum subsystems to other unmeasured quantum subsystems from the plurality of quantum systems; and a classical computing system for determining the result of a logic qubit gate based on the batch measurement result data.

[0012]

[0012] According to some embodiments, the system includes a first input coupled to a first qubit and a first switch, the first switch including a first output, a second output, and a third output. The system further includes a first single-qubit measurement device coupled to the first output of the first switch and a second single-qubit measurement device coupled to the first output of a second switch. The system further includes a first two-qubit measurement device coupled to the second output of the first switch and the second output of the second switch, and a second two-qubit measurement device coupled to the third output of the first switch and the third output of the second switch.

[0013]

[0013] In some embodiments, the system further includes a fused network controller circuit coupled to a first switch and a second switch.

[0014]

[0014] In some embodiments, the system further includes a decoder coupled to the output of a first single-qubit measurement device, the output of a second single-qubit measurement device, the output of a first two-qubit measurement device, and the output of a second two-qubit measurement device.

[0015]

[0015] In some embodiments, the first qubit is entangled with one or more other qubits as part of a first resource state, the second qubit is entangled with one or more other qubits as part of a second resource state, and none of the qubits from the first resource state are entangled with any of the qubits from the second resource state.

[0016]

[0016] In some embodiments, the first and second two-qubit measurement devices are configured to perform destructive batch measurements on the first and second qubits and to output classical information representing the batch measurement results.

[0017]

[0017] In some embodiments, the first qubit and the second qubit are photonic qubits.

[0018]

[0018] In some embodiments, the coupling between the first and second qubits and the first and second switches includes a plurality of photonic waveguides.

[0019]

[0019] In some embodiments, the first single-qubit measurement device is configured to measure the first qubit in the Z basis.

[0020]

[0020] In some embodiments, the second single-qubit measurement device is configured to measure the second qubit in the Z basis.

[0021]

[0021] In some embodiments, the first two-qubit measurement device is configured to perform a projected Bell measurement between the first qubit and the second qubit.

[0022]

[0022] In some embodiments, the second two-qubit measurement device is configured to perform a projected Bell measurement between the first qubit and the second qubit.

[0023]

[0023] In some embodiments, the projected Bell measurement is a linear optical type II fused measurement.

[0024]

[0024] In some embodiments, the projected Bell measurement is a linear optical type II fused measurement.

[0025]

[0025] The following detailed description, along with the accompanying drawings, will provide a better understanding of the nature and merits of the invention described in the claims.

[0026]

[0026] The embodiments of this disclosure are shown as examples. Non-limiting and non-exclusive embodiments are described in relation to the following figures, and unless otherwise specified, the same reference numbers throughout the various figures refer to the same parts. [Brief explanation of the drawing]

[0027] [Figure 1A]This figure shows cluster states and corresponding syndrome graphs related to the entanglement state of physical qubits according to several embodiments. [Figure 1B] This figure shows cluster states and corresponding syndrome graphs related to the entanglement state of physical qubits according to several embodiments. [Figure 1C] This figure shows cluster states and corresponding syndrome graphs related to the entanglement state of physical qubits according to several embodiments. [Figure 2] One or more quantum computing systems according to one embodiment are shown. [Figure 3A] Several quantum computing systems according to the present invention are shown. [Figure 3B] Several quantum computing systems according to the present invention are shown. [Figure 3C] Several quantum computing systems according to the present invention are shown. [Figure 3D] Several quantum computing systems according to the present invention are shown. [Figure 4A] An example of a fusion network according to several embodiments is shown. [Figure 4B] Several quantum computing systems according to the present invention are shown. [Figure 5] Several examples of qubit fusion systems according to certain embodiments are shown. [Figure 6] The resource states according to several embodiments are shown. [Figure 7] Several examples of qubit fusion systems according to certain embodiments are shown. [Figure 8A] An example of a fused network is shown, along with explicit definitions of specific groups according to several embodiments. [Figure 8B] An example of a fused network is shown, along with explicit definitions of specific groups according to several embodiments. [Figure 8C] An example of a fused network is shown, along with explicit definitions of specific groups according to several embodiments. [Figure 9A]An example of a fused network is shown, along with explicit definitions of specific groups according to several embodiments. [Figure 9B] An example of a fused network is shown, along with explicit definitions of specific groups according to several embodiments. [Figure 9C] An example of a fused network is shown, along with explicit definitions of specific groups according to several embodiments. [Figure 9D] An example of a fused network is shown, along with explicit definitions of specific groups according to several embodiments. [Figure 9E] An example of a fused network is shown, along with explicit definitions of specific groups according to several embodiments. [Figure 10A] An example of a fusion network according to several embodiments is shown. [Figure 10B] An example of a fusion network according to several embodiments is shown. [Figure 10C] An example of a fusion network according to several embodiments is shown. [Figure 10D] An example of a fusion network according to several embodiments is shown. [Figure 10E] An example of a fusion network according to several embodiments is shown. [Figure 11] This shows numerically calculated error tolerances for various fused network and resource states according to several embodiments. [Figure 12] This shows numerically calculated error tolerances for various fused network and resource states according to several embodiments. [Figure 13A] A simple example is shown of how primal and dual boundaries can be created by measuring specific qubits in a Z basis according to several embodiments. [Figure 13B] A simple example is shown of how primal and dual boundaries can be created by measuring specific qubits in a Z basis according to several embodiments. [Figure 13C]A simple example is shown of how primal and dual boundaries can be created by measuring specific qubits in a Z basis according to several embodiments. [Figure 13D] A simple example is shown of how primal and dual boundaries can be created by measuring specific qubits in a Z basis according to several embodiments. [Figure 14A] An example of a fusion router that provides routing to generate a 6-ring fusion network using a photonic resource state generator, optical routing, and linear optical fusion according to several embodiments is shown. [Figure 14B] An example of a fusion router that provides routing to generate a 6-ring fusion network using a photonic resource state generator, optical routing, and linear optical fusion according to several embodiments is shown. [Figure 14C] An example of a fusion router that provides routing to generate a 6-ring fusion network using a photonic resource state generator, optical routing, and linear optical fusion according to several embodiments is shown. [Figure 14D] An example of a fusion router that provides routing to generate a 6-ring fusion network using a photonic resource state generator, optical routing, and linear optical fusion according to several embodiments is shown. [Figure 14E] An example of a fusion router that provides routing to generate a 6-ring fusion network using a photonic resource state generator, optical routing, and linear optical fusion according to several embodiments is shown. [Figure 15A] For example, other embodiments of networked RSG circuits that can be used in material-based qubit architectures such as trapped ions and superconducting qubits according to several embodiments are shown. [Figure 15B] For example, other embodiments of networked RSG circuits that can be used in material-based qubit architectures such as trapped ions and superconducting qubits according to several embodiments are shown. [Figure 16]An example of a quantum circuit using logic feedforward according to several embodiments is shown. [Figure 17] The diagrams show the flow of classical information into and out of quantum computing systems according to several embodiments. [Figure 18] An example of a decoding system configuration that includes both buffering and decoder parallelization according to several embodiments is shown. [Figure 19] Several embodiments of photonic hardware components within a linear optical quantum computer are shown. [Figure 20] An example of a multiplexed single-photon source according to one or more embodiments is shown. [Figure 21] One possible example of a fusion site configured to work with a fusion controller to provide measurement results to a decoder for fault-tolerant quantum computing according to several embodiments is shown. [Figure 22A] This document presents one or more fusion-based quantum computing schemes for fault-tolerant quantum computing according to one or more embodiments. [Figure 22B] This document presents one or more fusion-based quantum computing schemes for fault-tolerant quantum computing according to one or more embodiments. [Figure 22C] This document presents one or more fusion-based quantum computing schemes for fault-tolerant quantum computing according to one or more embodiments. [Figure 23A] Several examples of lattice preprocessing protocols for fusion-based quantum computing according to certain embodiments are shown. [Figure 23B] Several examples of lattice preprocessing protocols for fusion-based quantum computing according to certain embodiments are shown. [Figure 23C] Several examples of lattice preprocessing protocols for fusion-based quantum computing according to certain embodiments are shown. [Figure 24A] Several examples of lattice preprocessing protocols for fusion-based quantum computing according to certain embodiments are shown. [Figure 24B]Several examples of lattice preprocessing protocols for fusion-based quantum computing according to certain embodiments are shown. [Figure 25A] A flowchart and exemplary lattice preprocessing protocol are shown to illustrate a method for fusion-based quantum computing according to one or more embodiments. [Figure 25B] A flowchart and exemplary lattice preprocessing protocol are shown to illustrate a method for fusion-based quantum computing according to one or more embodiments. [Figure 25C] A flowchart and exemplary lattice preprocessing protocol are shown to illustrate a method for fusion-based quantum computing according to one or more embodiments. [Figure 25D] A flowchart and exemplary lattice preprocessing protocol are shown to illustrate a method for fusion-based quantum computing according to one or more embodiments. [Figure 25E] A flowchart and exemplary lattice preprocessing protocol are shown to illustrate a method for fusion-based quantum computing according to one or more embodiments. [Figure 26A] The diagrams show dual-rail coded photonic qubits and photonic circuits for performing unitary operations on photonic qubits according to several embodiments. [Figure 26B] The diagrams show dual-rail coded photonic qubits and photonic circuits for performing unitary operations on photonic qubits according to several embodiments. [Figure 26C] The diagrams show dual-rail coded photonic qubits and photonic circuits for performing unitary operations on photonic qubits according to several embodiments. [Figure 26D] The diagrams show dual-rail coded photonic qubits and photonic circuits for performing unitary operations on photonic qubits according to several embodiments. [Figure 26E]The diagrams show dual-rail coded photonic qubits and photonic circuits for performing unitary operations on photonic qubits according to several embodiments. [Figure 27A] The diagrams show dual-rail coded photonic qubits and photonic circuits for performing unitary operations on photonic qubits according to several embodiments. [Figure 27B] The diagrams show dual-rail coded photonic qubits and photonic circuits for performing unitary operations on photonic qubits according to several embodiments. [Figure 28] This document describes a photonic implementation of a beam splitter that can be used to implement one or more spreaders, such as Hadamard gates, according to several embodiments. [Figure 29] This document describes a photonic implementation of a beam splitter that can be used to implement one or more spreaders, such as Hadamard gates, according to several embodiments. [Figure 30] An example of a Bell state generator circuit that can be used in several dual-rail coding photonic embodiments is shown. [Figure 31] An example of a Type II fusion circuit for polarization coding according to several embodiments is shown. [Figure 32] An example of a Type II fusion circuit for path coding according to several embodiments is shown. [Figure 33A] The effects of fusion in the generation of cluster states according to several embodiments are shown. [Figure 33B] The effects of fusion in the generation of cluster states according to several embodiments are shown. [Figure 33C] The effects of fusion in the generation of cluster states according to several embodiments are shown. [Figure 33D] The effects of fusion in the generation of cluster states according to several embodiments are shown. [Figure 34]Examples of once-boosted Type II fusion gates in polarization coding and path coding according to several embodiments are shown. [Figure 35] A table is shown showing variations of type II fusion gates for different measurement bases in polarization coding. [Figure 36] Examples of photonic circuit variations of type II fusion gates for different selections of measurement basis in path coding according to several embodiments are shown. [Modes for carrying out the invention]

[0028]

[0064] Herein, embodiments are given in detail, examples of which are shown in the accompanying drawings. In the following detailed description, numerous specific details are given in order to give a complete understanding of the various described embodiments. However, as will be apparent to those skilled in the art, the various described embodiments may be carried out without these specific details. In other examples, well-known methods, procedures, components, circuits, and networks are not described in detail so as not to unnecessarily obscure the aspects of the embodiments.

[0029]

[0065] I. Introduction to Quantum Computing Quantum computing is often considered within the framework of "circuit-based quantum computing" (CBQC), where operations (or gates) are performed on physical qubits. Gates can be single-qubit unitary operations (rotations), two-qubit entanglement operations such as CNOT gates, or other multi-qubit gates such as Toffoli gates.

[0030]

[0066] Measurement-based quantum computing (MBQC) is another method for performing quantum computation. In the MBQC method, computation proceeds by first pre-processing a specific entangled state of many qubits, commonly called a cluster state, and then performing a quantum computation by performing a series of single-qubit measurements on the cluster state. In this method, the selection of single-qubit measurements is determined by the quantum algorithm being run on the quantum computer. Fault tolerance can be achieved in the MBQC method by carefully designing the cluster state and using the topology of this cluster state to encode logical qubits that are protected from any logical errors that may be caused by errors in any of the physical qubits that make up the cluster state. In practice, the values ​​of the logical qubits can be determined, or read out, based on the results of single-particle measurements (also referred to herein as measurement results) performed on the physical qubits of the cluster state as the computation progresses.

[0031]

[0067] However, generating and maintaining long-range entanglement across cluster states, and subsequently storing large cluster states, can be challenging. For example, in any physical embodiment of the MBQC method, cluster states containing thousands or more entangled qubits must be preprocessed and then stored for a period before single-qubit measurements are performed. For instance, to generate a cluster state representing a single logical error-corrected qubit, each of the underlying set of physical qubits can be preprocessed into a |+〉 state, and a controlled phase gate (CZ) can be applied between each pair of physical qubits to generate the entire cluster state. More explicitly, a cluster state of highly entangled qubits can be described by an undirected graph G=(V,E) where V and E represent sets of vertices and edges, respectively, and can be generated as follows: i.e., 1) initialize all physical qubits into a |+〉 state, where,

number

number

[0032]

[0068] |Ψ〉 graph After the generation of these entangled qubits, the large state of these entangled qubits must be preserved long enough for stabilizer measurements to be performed, for example, by performing X measurements on most of the physical qubits in the lattice and Z measurements on the boundary qubits.

[0033]

[0069] Figure 1A shows an example of a fault-tolerant cluster state that can be used in MBQC, the topological cluster state being called a Raussendorf lattice, as introduced by Raussendorf et al. and described in more detail in Robert Raussendorf, Jim Harrington, and Kovid Goyal.A., Fault-Tolerant One-Way Quantum Computer, Annals of Physics, 321(9):2242-2270, 2006. The cluster state is in the form of a repeating lattice cell (e.g., cell 120) having physical qubits (e.g., physical qubit 116) arranged on the faces and edges of the cell. The entanglement between physical qubits is represented by edges connecting the physical qubits (e.g., edge 118), each edge representing the application of a CZ gate as described above with reference to equation (1). The cluster state shown herein is merely one example among many other topological error correction codes that can be used without departing from the scope of this disclosure. For example, volume codes such as those disclosed in International Patent Application Publication 2019 / 173651, whose entire contents are incorporated herein by reference for all purposes, can be used. Similarly, codes based on non-cubic unit cells, as described in International Patent Application Publication 2019 / 178009, whose entire contents are incorporated herein by reference for all purposes, can also be used without departing from the scope of this disclosure. Furthermore, although the examples shown herein are represented in three spatial dimensions, the same structure may be obtained from other implementations of codes that are not based purely on spatially entangled cluster states, but rather can include both entanglement in 2D space and entanglement in time, for example, a 2+1D surface code implementation may be used, or any other fragmented code may be used. For cluster state implementations of such codes, all the quantum gates necessary for fault-tolerant quantum computing can be constructed by performing a series of single-particle measurements on the physical qubits constituting the lattice.

[0034]

[0070] Returning to Figure 1A, a chunk of the Raussendorf lattice is shown. Such an entangled state can be used to encode one or more logical qubits (i.e., one or more error-corrected qubits) using many entangled physical qubits. A set of single-particle measurement results of multiple physical qubits (e.g., 116 physical qubits) can be used to correct errors and perform fault-tolerant calculations on the logical qubits using a decoder. Many decoders are available, one example being the Union-Find decoder described in International Patent Application Publication No. 2019 / 002934A1, the disclosures of which are incorporated herein by reference in their entirety for all purposes. As those skilled in the art will see, the number of physical qubits required to encode a single logical qubit may vary depending on the exact nature of the physical errors, noise, etc., experienced by the physical qubits, but to achieve fault tolerance, all previous proposals require an entangled state of thousands of physical qubits to encode a single logical qubit. Generating and maintaining such large entanglements remains a significant challenge for any practical implementation of MBQC methods.

[0035]

[0071] Figures 1B and 1C illustrate how the decoding of logical qubits can proceed for cluster states based on a Raussendorf lattice. As seen in Figure 1A, the geometric shape of the cluster state is related to the geometric shape of the cubic lattice (lattice cell 120) superimposed on the cluster state in Figure 1A. Figure 1B shows the single-particle measurement results (superimposed on the cubic lattice) after the state of each physical qubit in the cluster state has been measured, with the measurement results placed in the previous position of the measured physical qubit (for clarity, only measurement results resulting from surface qubit measurements are shown).

[0036]

[0072] In some embodiments, the measured qubit state can be represented by a numerical bit value of either 1 or 0 after all qubits have been measured, for example, in an x ​​basis, where a bit value of 1 corresponds to a +x measurement result, and a value of 0 corresponds to a -x measurement result (or vice versa). There are two types of qubits: those located on the edges of the unit cell (e.g., in edge qubit 122) and those located on the faces of the unit cell (e.g., face qubit 124). In some cases, a qubit measurement may not be available, or the measurement result of a qubit may be invalid. In these cases, there is no bit value assigned to the corresponding measured qubit position; instead, the result is called erasure, shown herein, for example, as bold line 126. These measurement results, which are known to be missing, can be reconstructed during the decoding procedure.

[0037]

[0073] To identify errors in physical qubits, a syndrome graph can be generated from the set of measurement results resulting from physical qubit measurements. For example, the bit values ​​associated with each edge qubit can be combined to create a syndrome value associated with a vertex resulting from the intersection of each edge, e.g., vertex 128 as shown in Figure 1B. The set of syndrome values, also referred to herein as parity checks, is associated with each vertex of the syndrome graph, as shown in Figure 1C. More specifically, Figure 1C shows the calculated values ​​of parity checks for some vertex parities of the syndrome graph. In some embodiments, the parity calculation involves determining whether the sum of the edge values ​​incident on a given vertex is an even or odd integer, and the parity result for that vertex is defined as the result of the sum modulo 2. If no errors occur in the quantum state or qubit measurement, all syndrome values ​​should be even (or 0). Conversely, if an error occurs, it results in several odd (or 1) syndrome values. Only half of the bit values ​​from the qubit measurement are associated with the shown syndrome graph (bits aligned to the edges of the syndrome graph). There is another syndrome graph containing all the bit values ​​associated with the faces of the grid shown. This brings about an equivalent decoding problem for these bits.

[0038]

[0074] As mentioned above, generating and subsequently storing large cluster states of qubits can be a challenge. However, some embodiments, methods, and systems described herein provide the generation of a set of classical measurement data (e.g., a set of classical data corresponding to syndrome graph values ​​in a syndrome graph) containing the correlations necessary to perform quantum error correction without first generating large entangled states of qubits in an error correction code. For example, embodiments disclosed herein describe a system and method on which a set of classical data containing long-range correlations necessary to generate and decode a syndrome graph of a particular selected cluster state can be generated by performing a two-qubit (i.e., junction) measurement, also referred herein as a “fusion measurement” or “fusion gate,” on a much smaller set of entangled states, without actually generating the cluster states. In other words, in some of the systems and methods described herein, only a set of relatively small entangled states (referred to herein as resource states) is generated, and then, without the need to first generate (and then measure) large cluster states that form quantum error correction codes (e.g., topological codes such as Raussendorf lattices), a batch measurement is performed directly on these resource states to generate syndrome graph data.

[0039]

[0075] For example, in the case of linear optical quantum computing that uses a Raussendorf lattice coding structure to generate syndrome graph data, as will be explained in more detail below, the fusion gates can be applied to a set of small entangled states (e.g., 4-GHz states) that are not entangled with each other and therefore never become part of a larger Raussendorf lattice cluster state. Despite the fact that the qubits from the individual resource states were not entangled with each other before the fusion measurement, the batch measurement result resulting from the fusion measurement generates a syndrome graph containing all the correlations necessary to perform quantum error correction. Such systems and methods are referred to herein as fusion-based quantum computing (FBQC). Preferably, the resource states have a size independent of the computation performed or the code distance used, in stark contrast to the cluster states in MBQC. This allows the resource states used in FBQC to be generated by a fixed number of sequential operations. As a result, in FBQC, errors in the resource states are limited, which is important for fault tolerance.

[0040]

[0076] II. Systems for FBQC Figure 2 shows a quantum computing system according to one or more embodiments. The quantum computing system 201 includes a user interface device 204 that is communicatively coupled to a quantum computing (QC) subsystem 206, which is described in more detail below in Figure 3. The user interface device 204 can be any type of user interface device, such as a terminal including a display, keyboard, mouse, touchscreen, etc. Furthermore, the user interface device itself can be a computer such as a personal computer (PC), laptop, or tablet computer. In some embodiments, the user interface device 204 provides an interface that allows a user to interact with the QC subsystem 206 directly, via a local area network, a wide area network, or via the Internet. For example, the user interface device 204 can run software such as a text editor, an interactive development environment (IDE), a command prompt, or a graphical user interface, thereby allowing the user to program the QC subsystem to execute one or more quantum algorithms or otherwise interact with the QC subsystem. In other embodiments, the QC subsystem 206 may be pre-programmed, and the user interface device 204 may simply be an interface from which the user can initiate quantum computing, monitor its progress, and receive results from the QC subsystem 206. The QC subsystem 206 further includes a classical computing system 208 coupled to one or more quantum computing chips 210. In some examples, the classical computing system 208 and the quantum computing chips 210 can be coupled to other electronic components 212, such as pulse pump lasers, microwave oscillators, power supplies, network hardware, etc. In some embodiments using cryogenic operation, the quantum computing system 201 can be housed in a cryostat, such as a cryostat 214.In some embodiments, the quantum computing chip 210 may include an integration (direct or heterogeneous) of one or more component chips, for example, an electronic chip 216 and an integrated photonics chip 218. Signals may be routed on-chip and off-chip in any number of ways, for example, via optical interconnects 220 and other electronic interconnects 222. Furthermore, the computing system 201 may use quantum computing processes, for example, fusion-based quantum computing processes, such as those described in more detail below.

[0041]

[0077] Figure 3A shows a block diagram of a QC system 301 according to several embodiments. Such a system can be related to the computing system 201 described above with reference to Figure 2. In Figure 3, solid lines represent quantum information channels, and double solid lines represent classical information channels. The QC system 301 includes a resource state generator 303, a qubit fusion system 305, and a classical computing system 307. In some embodiments, the resource state generator 303 can take as input a set of N physical qubits (also referred to herein as "quantum subsystems"), for example, physical qubits 309 (also schematically represented as inputs 311a, 311b, 311c, ..., 311N), and can generate quantum entanglement between two or more of them to generate an entangled resource state 315 (also referred to as a "quantum system" which itself consists of entangled states of quantum subsystems). For example, in the case of photonic qubits, the resource state generator 303 can be a linear optical system such as an integrated photonic circuit including waveguides, beam splitters, photon detectors, delay lines, etc. In some examples, the entangled resource states 315 may be relatively small entangled states of qubits (e.g., qubit entangled states having 3 to 30 qubits). In some embodiments, the resource states can be selected such that a fusion operation applied to specific qubits in these states yields syndrome graph data containing the correlations necessary for quantum error correction. Preferably, the system shown in Figure 3 provides fault-tolerant quantum computing using relatively small resource states without requiring the resource states to be entangled with each other to form typical lattice cluster states required for MBQC.

[0042]

[0078] In some embodiments, the input qubit 309 can be a collection of quantum systems (also referred herein as quantum subsystems) and / or particles, which can be formed using any qubit architecture. For example, a quantum system can be a particle such as an atom, ion, nucleus, and / or photon. In other examples, a quantum system can be a flux qubit, a phase qubit, or a charge qubit (e.g., formed from a superconducting Josephson junction), a topological qubit (e.g., a majorana fermions), a spin qubit formed from a vacancy center (e.g., a nitrogen vacancy in diamond), or a qubit otherwise encoded in a multiple quantum system, such as a Gottesman-Kitaev-Preskill (GKP) encoded qubit. Furthermore, for the sake of clarity, the term “qubit” is used herein, but the system can also use quantum information carriers that encode information in a manner not necessarily associated with binary bits. For example, according to some embodiments, a qubit (i.e., a quantum system capable of encoding information in three or more quantum states) can be used.

[0043]

[0079] According to several embodiments, the QC system 301 can be a fusion-based quantum computer capable of executing one or more quantum algorithms or software programs. For example, a software program representing a quantum algorithm to be executed on the QC system 301 (e.g., a set of machine-readable instructions) can be passed to a classical computing system 307 (e.g., corresponding to system 208 in Figure 2 above). The classical computing system 307 may be any type of computing device, such as a PC, one or more blade servers, or a high-performance computing system, such as a supercomputer or server farm. Such a system may include one or more processors (not shown) coupled to one or more computer memories, such as memory 306. Such a computing system is referred to herein as a “classical computer”. In some examples, a logic processor 308 can receive a software program as input and compute a corresponding set of logic gates to be applied to execute the software program on specific hardware available within the QC system 301. In some embodiments, the software program can be received by other, or even more, modules, such as a fusion pattern generator 313. One function of the fusion pattern generator 313 is to generate a set of machine-level fusion instructions (e.g., a set of fusion operations and / or single-qubit measurements applied across the physical qubits constituting the QC system 301). Thus, the logic processor 308 and the fusion pattern generator 313 can receive an input software program (which can be generated as a high-level code that can be more easily written by a user to program a quantum computer) and generate a set of machine-readable instructions to be applied to low-level quantum hardware.

[0044]

[0080] In some embodiments, the fusion pattern generator 313 can operate (either alone or in combination with the logic processor 308) as a compiler for software programs executed on the quantum computer. The fusion pattern generator 313 can be implemented as pure hardware, pure software, or any combination of one or more hardware or software components or modules. In various embodiments, the fusion pattern generator 313 can operate at runtime or in advance, and in either case, the machine-level instructions generated by the fusion pattern generator 313 can be stored (e.g., in memory 306). In some examples, the compiled machine-level instructions take the form of one or more data frames that instruct the qubit fusion system 305 to perform one or more fusions between specific qubits from separate, i.e., unentangled resource states 315 in a given clock cycle of the quantum computer. For example, a fusion pattern data frame 317 is an example of a set of fusion measurements (e.g., Type II fusion measurements, described in more detail below with reference to Figures 18-21) to be applied between specific pairs of qubits from different entangled resource states 315 during a particular clock cycle when the program is executed.

[0045]

[0081] In some embodiments, several fusion pattern data frames 317 can be stored in memory 306 as classical data. In some embodiments, the fusion pattern data frame 317 can indicate whether an XX type II fusion is applied to a particular fusion gate in the fusion array 321 of the qubit fusion system 305 (or whether any other type of fusion is applied). Furthermore, the fusion pattern data frame 317 can indicate that a type II fusion is performed on different bases, e.g., an XX fusion, an XY fusion, a ZZ fusion, etc. As used herein, terms such as XX type II fusion, YY type II fusion, XY type II fusion, and ZZ type II fusion refer to a fusion operation that applies a particular two-particle projection measurement, e.g., a Bell projection that can project two qubits onto one of four Bell states depending on a selected Bell base. Such a projection measurement produces two measurement results (also referred to herein as batch measurement result data) corresponding to the corresponding pair of eigenvalues ​​of the observable event measured on the selected base. For example, XX fusion is a Bell projection that measures the XX and ZZ observables (each having eigenvalues ​​of +1 or -1, or 0 or 1 depending on the convention used), XZ fusion is a Bell projection that measures the observable XZ and ZX observables, and so on. Figures 23 to 36 below show exemplary circuits for performing type II fusion for various base selections in a linear optical system, but other Bell projection measurements are possible in other qubit architectures without departing from the scope of this disclosure. As those skilled in the art will see, in a linear optical system, type II fusion performs a stochastic Bell measurement. Figures 23 to 36 illustrate the stochastic nature of linear optical fusion in relation to fusion “success” and “failure” outcomes, which are not repeated here for clarity.

[0046]

[0082] The fusion network controller circuit 319 of the qubit fusion system 205 can receive data encoding a fusion pattern data frame 317, and based on this data can generate configuration signals, such as analog and / or digital electronic signals, to drive hardware in the fusion array 321. For example, in the case of photonic qubits, the fusion gate may include a photon detector coupled to one or more waveguides, beam splitters, interferometers, switches, polarizers, polarization rotors, etc. More generally, the detector can be any detector capable of detecting one or more quantum states of the qubits in the resource state 315. As those skilled in the art will see, many types of detectors can be used depending on the specific qubit architecture used.

[0047]

[0083] In some embodiments, the result of applying the fusion pattern data frame 317 to the fusion array 321 is the generation of classical data (generated by the detectors of the fusion gates) that is read out directly (not shown) or via any other module, optionally preprocessed, and sent to the fusion pattern generator and / or decoder 333. More specifically, the fusion array 321 (also referred herein as the “fusion network”) may include a set of measurement devices that perform batch measurements between specific qubits from two different resource states and generate a set of measurement results associated with the batch measurements. These measurement results (also referred herein as batch measurement result data) can be stored in a measurement result data frame, e.g., data frame 322, and returned to a classical computing system for further processing. In some embodiments, by passing the measurement result data frame 322 directly to the fusion pattern generator, a rapid adaptive feedforward process can be enabled that allows the system to modify the fusion pattern data frame 317 in a future clock cycle (e.g., a selection of a basis or a selection of a single-particle measurement) based on the measurement result data collected in a previous time step.

[0048]

[0084] In some embodiments, any of the submodules within the QC system 301, such as the controller 323, quantum gate array 325, fusion array 321, fusion network controller 319, fusion pattern generator 313, decoder 333, and logic processor 308, may include any number of classical computing components such as processors (CPU, GPU, TPU), memory (any form of RAM, ROM), hard-coded logic components (classical logic gates such as AND, OR, XOR), and / or programmable logic components such as field-programmable gate arrays (FPGA, etc.). These modules may also include any number of application-specific integrated circuits (ASICs), microcontrollers (MCUs), systems-on-chip (SOCs), and other similar microelectronics. Figure 3 shows a particular set of modules exchanging data, signals, and messages to perform functions as described above, but as those skilled in the art will see, the particular arrangement of modules shown herein is merely an example, and many different examples are possible without departing from the scope of this disclosure. For example, the compilation, feedforward functions described above can be shared between modules.

[0049]

[0085] In some embodiments, the entangled resource state 315 can be any type of entangled resource state that, when a fusion operation is performed, generates a measurement results data frame containing the correlations necessary to perform fault-tolerant quantum computation. Figure 3 shows an example of a set of identical resource states, but a system can be employed that can generate many different types of resource states and even dynamically change the type of resource state being generated based on the requirements of the quantum algorithm being executed. As described herein, the logical qubit measurement result 327 can be fault-tolerantly recovered from the physical qubit measurement result 322, for example, via a decoder 333. The logic processor 308 can then process the logic result as part of program execution. As shown, the logic processor can feed back information to the fusion pattern generator 313 to influence downstream gates and / or measurements to ensure that the computation proceeds fault-tolerantly.

[0050]

[0086] Figure 3B shows an example of a resource state generator 401 according to several embodiments. Such a system can be used, according to several embodiments, to generate qubits (e.g., photons) in entangled states (e.g., resource states used in the exemplary examples shown in Figures 7 to 9 below). The resource state generator 401 is an example of a system that can be employed in an FBQC system, such as the resource state generator 303 shown in Figure 3 above. As those skilled in the art will see, any resource state generator can be used without departing from the scope of this disclosure. Examples of resource state generators can be found in U.S. Patent Application No. 16 / 621,994, titled "Generation of entangled qubit states" (published as U.S. Patent Publication No. 20200287631), U.S. Patent Application No. 16 / 691,459, titled "GENERATION OF ENTANGLED PHOTONIC STATES" (published as U.S. Patent No. 11,126,062), and U.S. Patent Application No. 16 / 691,450, titled "GENERATION OF ENTANGLED PHOTONIC STATE FROM PRIMITIVE RESOURCES" (published as U.S. Patent Publication No. XXXXX), the disclosures of which are incorporated herein by reference in their entirety for all purposes. For example, in some embodiments, instead of generating a single photon, the photon source may directly generate an entangled resource state, or further, it may generate a smaller entangled state that can undergo additional entanglement in the entanglement state generator 400 to generate the final resource state used for FBQC. Thus, as used herein, the scope of the term “photon source” is intended to include at least a single-photon source, a source of multiple photons in an entangled state, or more generally, any source of a photonic state. As those skilled in the art will see, the exact form of the resource state generation hardware is not important, and any system can be used without departing from the scope of this disclosure.

[0051]

[0087] In an exemplary photonic architecture, the resource state generator 401 may include a photon source system 405 optically connected to the entanglement state generator 400. Both the photon source system 405 and the entanglement state generator 400 may be coupled to a classical processing system 403 so that the classical processing system 403 can communicate with and / or control the photon source system 405 and / or the entanglement state generator 400 (e.g., via classical information channels 430a-b). The photon source system 405 may include a collection of single-photon sources that can provide output photon states (e.g., single photons, or other photonic states such as Bell states, GHz states, etc.) to the entanglement state generator 400 by interconnecting waveguides 402. The entangled state generator 400 can receive output photonic states, convert them into one or more entangled photonic states (or a larger photonic state if the source itself outputs entangled photonic states), and then output these entangled photonic states to the output waveguide 440. In some embodiments, the output waveguide 440 can be coupled to some downstream circuit that can use the entangled states to perform quantum computations. For example, the entangled states generated by the entangled state generator 400 can be used as a resource for a downstream quantum optical circuit (not shown).

[0052]

[0088] In some embodiments, the photon source system 405 and the entangled state generator 400 can be used in conjunction with the quantum computing system shown in Figure 3. For example, the resource state generator 303 shown in Figure 3 may include the photon source system 405 and the entangled state generator 400, and the classical computer system 403 in Figure 4 may include one or more of the various classical computing components shown in Figure 3 (e.g., classical computing system 307). In this case, the entangled photons exiting through the output waveguide 440 can be fused together by the qubit fusion system 305, i.e., input into a detection system that performs a batch measurement set for use in the FBQC scheme.

[0053]

[0089] In some embodiments, the system 401 may include classical channels 430 (e.g., classical channels 430-a to 430-d) for interconnecting and providing classical information between components. It should be noted that classical channels 430-a to 430-d do not all need to be the same. For example, classical channels 430-a to 430-c may comprise a bidirectional communication bus that carries one or more reference signals, e.g., one or more clock signals, one or more control signals, or any other signals that carry classical information, e.g., herald signals, photon detector readout signals, etc.

[0054]

[0090] In some embodiments, the resource state generator 401 includes a classical computer system 403 that communicates with and / or controls the photon source system 405 and / or the entangled state generator 400. For example, in some embodiments, the classical computer system 403 can be used to configure one or more circuits, for example, a system clock that can be provided to the photon source 405 and the entangled state generator 400, and any downstream quantum photonic circuits used to perform quantum computations. In some embodiments, the quantum photonic circuits may include optical circuits, electrical circuits, or any other type of circuit. In some embodiments, the classical computer system 403 includes a memory 404, one or more processors 402, a power supply, an input / output (I / O) subsystem, and a communication bus or interconnecting these components. The processors 402 can execute software modules, programs, and / or instructions stored in the memory 404, thereby performing processing operations.

[0055]

[0091] In some embodiments, memory 404 stores one or more programs (e.g., sets of instructions) and / or data structures. For example, in some embodiments, the entanglement state generator 400 may attempt to generate entanglement states across successive stages and / or independent instances, any one of which may succeed in generating an entanglement state. In some embodiments, memory 404 stores one or more programs for determining whether each stage is successful and configuring the entanglement state generator 400 accordingly (e.g., by configuring the entanglement state generator 400 to switch a photon to output if the stage is successful, or to pass a photon to the next stage of the entanglement state generator 400 if the stage has not yet succeeded). To this end, in some embodiments, memory 404 stores detection patterns that a classical computing system 403 can use to determine whether a stage is successful. Furthermore, memory 404 may store settings provided to various configurable components described herein (e.g., switches), which are configured, for example, by setting one or more phase shifts of components.

[0056]

[0092] In some embodiments, some or all of the functions described above can be implemented by hardware circuits on the photon source system 405 and / or the entanglement state generator 400. For example, in some embodiments, the photon source system 405 includes one or more controllers 407-a (e.g., logic controllers) (e.g., a field-programmable gate array (FPGA), an application-specific integrated circuit (ASICS), a "system on a chip" including a classical processor and memory, etc.). In some embodiments, the controller 407-a determines whether the photon source system 405 has succeeded (e.g., for a given attempt for a given clock cycle) and outputs a reference signal indicating whether the photon source system 405 has succeeded. For example, in some embodiments, the controller 407-a outputs a logic high to classical channels 430-a and / or classical channels 430-c if the photon source system 405 has succeeded, and outputs a logic low to classical channels 430-a and / or classical channels 430-c if the photon source system 405 has failed. In some embodiments, the output of controller 407-a may be used to configure hardware within controller 107-b.

[0057]

[0093] Similarly, in some embodiments, the entanglement state generator 400 includes one or more controllers 407-b (e.g., logic controllers) (e.g., field-programmable gate arrays (FPGAs), application-specific integrated circuits (ASICS), etc.) that determine whether each stage of the entanglement state generator 400 has succeeded, execute the aforementioned switching logic, and output reference signals to classical channels 430-b and / or 430-d to notify other components of whether the entanglement state generator 400 has succeeded.

[0058]

[0094] In some embodiments, a system clock signal may be provided to the photon source system 405 and the entanglement state generator 400 via an external source (not shown) or by a classical computing system 403 via classical channels 430-a and / or 430-b. Examples of clock generators that may be used are described in U.S. Patent No. 10,379,420, the entirety of which is incorporated herein by reference for all purposes, but other clock generators may also be used without departing from the scope of this disclosure. In some embodiments, the system clock signal provided to the photon source system 405 triggers the photon source system 405 to attempt to output one photon per waveguide. In some embodiments, the system clock signal provided to the entanglement state generator 400 triggers or gates a set of detectors in the entanglement state generator 400 to attempt to detect a photon. For example, in some embodiments, triggering a set of detectors in the entanglement state generator 400 to attempt to detect a photon includes gate-controlling the set of detectors.

[0059]

[0095] It should be noted that in some embodiments, the photon source system 405 and the entanglement state generator 400 may have internal clocks. For example, the photon source system 405 may have an internal clock generated and / or used by controller 407-a, and the entanglement state generator 400 may have an internal clock generated and / or used by controller 407-b. In some embodiments, the internal clock of the photon source system 405 and / or the entanglement state generator 400 are synchronized to an external clock (e.g., a system clock provided by a classical computer system 403) (e.g., via a phase-locked loop). In some embodiments, either of the internal clocks may be used as a system clock by itself; for example, the internal clock of the photon source may be distributed to other components in the system and used as a master / system clock.

[0060]

[0096] In some embodiments, the photon source system 405 includes a plurality of probabilistic photon sources that can be spatially and / or temporally multiplexed, i.e., so-called multiplexed single-photon sources. In one example of such a photon source, the photon source is driven by a pump, e.g., an optical pulse, coupled to an optical resonator capable of producing zero, one, or more photons by some nonlinear process (e.g., spontaneous four-wave mixing, second harmonic generation, etc.). As used herein, the term “try” is used to refer to the act of driving the photon source with some kind of drive signal, e.g., a pump pulse, which can produce output photons nondeterministically (i.e., the probability that the photon source will produce one or more photons in response to the drive signal may be less than 1). In some embodiments, each photon source may most likely produce zero photons in each try (e.g., the probability of producing zero photons per attempt to produce a single photon may be 90%). The second most likely outcome of the attempt is the production of a single photon (for example, the probability of producing a single photon for each attempt to produce a single photon may be 9%). The third most likely outcome of the attempt is the production of two photons (for example, the probability of producing two photons for each attempt to produce a single photon may be about 1%). In some circumstances, the probability of producing three or more photons may be less than 1%.

[0061]

[0097] In some embodiments, the apparent efficiency of a photon source can be increased by using multiple single-photon sources and multiplexing the outputs of multiple photon sources. In some embodiments, the photon source can also generate a classical herald signal to indicate (or signal) successful generation. In some embodiments, this classical signal is obtained from the output of a detector, the photon source system always generates photon states in pairs (e.g., SPDC), and the detection of a single photon signal is used to indicate success of the process. As will be described in more detail below, this herald signal may be provided to a multiplexer and used to properly route successful generation to the multiplexer output port.

[0062]

[0098] The exact type of photon source used is not important, and any type of source can be used with any photon generation process, such as spontaneous four-wave mixing (SPFW), spontaneous parametric down-conversion (SPDC), or any other process. Other classes of photon sources that do not necessarily require nonlinear materials can also be used, such as quantum dot sources, or those using atomic and / or artificial atomic systems such as color centers in crystals. In some cases, the photon source may or may not be coupled to a photonic cavity, for example, in the case of artificial atomic systems such as quantum dots coupled to a cavity. Other types of photon sources, such as optic-mechanical systems, also exist in SPWM and SPDC. In some examples, the photon source can emit multiple photons that are already in an entangled state, in which case the entanglement state generator 400 may not be necessary, or it may take an entangled state as input and generate an even larger entangled state.

[0063]

[0099] In some embodiments, spatial multiplexing of several nondeterministic photon sources (also known as MUX photon sources) can be used. Many different spatial MUX architectures are possible without departing from the scope of this disclosure. Temporal MUX can also be implemented instead of or in combination with spatial multiplexing. MUX schemes can be used that use logarithmic trees, generalized Mach-Zehnder interferometers, multimode interferometers, chain sources, dump-to-pump chain sources, asymmetric polycrystalline single-photon sources, or any other type of MUX architecture. In some embodiments, the photon source can be used in a MUX scheme that includes quantum feedback control, etc. An example of an n×mMUX source is disclosed in U.S. Patent No. 10,677,985, the contents of which are incorporated herein by reference in whole for any purpose.

[0064]

[0100] Figure 3C shows an example of a qubit fusion system 501 according to several embodiments. In some embodiments, the qubit fusion system 501 can be used within a larger FBQC system, such as the qubit fusion system 305 shown in Figure 3A.

[0065]

[0101] The qubit fusion system 501 includes a fusion network controller 519 coupled to a fusion array 521 (also referred to herein as the “fusion network”). The fusion network controller 519 is configured to operate as described above and below with reference to the fusion network controller circuit 319 in Figure 3 above. The fusion array 521 includes a set of fusion sites that each receive two or more qubits from different resource states (e.g., as shown in Figure 4a), perform one or more fusion operations (e.g., type II fusion) on qubits selected from two or more resource states, and / or implement fault-tolerant logic by performing selected single-particle measurements, as described below in more detail with reference to Figures 13-14. The measurement operations performed on the qubits can be controlled by the fusion network controller 519 via classical signals transmitted from the fusion network controller 519 to each of the fusion sites via control channels 503a, 503b, etc. Based on the measurements performed at each fusion site, classical measurement results in the form of classical data are output and provided to the decoder system as described above with reference to Figure 3. Examples of photonic circuits that can be used as Type II fusion gates are described below with reference to Figures 20 and 23-35.

[0066]

[0102] Figure 3D shows one possible example of a fusion region 341 (one of many constituting a fusion array 321) configured to work with a fusion network controller 319 to provide measurement results to a decoder system for fault-tolerant quantum computing according to several embodiments. In this example, the fusion region 341 can be an element of the fusion array 321 (shown in Figure 3), and only one example is shown for illustrative purposes, but the fusion array 321 can contain any number of examples of fusion region 341. In some embodiments, quantum logic gates can be implemented by modifying the fusion measurements. To enable the implementation of logic (at least), a subset of the fusion devices may be reconfigurable, as shown in Figure 3B, but others do not need to be reconfigurable. Bulk boundaries or other topological features can be implemented by modifying the fusion measurement basis, or by selecting single-qubit measurements instead of fusion, as also described below with reference to Figures 13A-13D.

[0067]

[0103] As previously mentioned, the qubit fusion system 305 can receive (with two or more inputs) two or more qubits (qubit 1 and qubit 2) to be measured according to the quantum application being performed. Qubit 1 incident on input 1 is one qubit entangled with one or more other qubits (not shown) as part of a first resource state, and qubit 2 incident on input 2 is another qubit entangled with one or more other qubits (not shown) as part of a second resource state. Preferably, in contrast to MBQC, to facilitate fault-tolerant quantum computation, none of the qubits from the first resource state need to be entangled with any of the qubits from the second (or any other) resource state. Also preferably, at the input of the fusion site 341, the set of resource states are not entangled with each other to form a cluster state in the form of a quantum error correction code, and therefore, there is no need to store and / or maintain a large cluster state with long-range entanglement across the entire cluster state. Preferably, in some embodiments, the fusion operation performed at the fusion site may be a completely destructive batch measurement on qubit 1 and qubit 2 such that all that remains after the measurement is classical information representing the measurement results on the detector, e.g., measurement results 603, 605, 607, 609, etc. At this point, the classical information is all that the decoder 333 needs to perform quantum error correction. This may be in contrast to MBQC systems that use a fusion site to fuse resource states into cluster states that themselves function as topological codes, and only then generate the necessary classical information through the additional step of single-particle measurements for each qubit in the large cluster state. In such MBQC systems, not only is it necessary to store and maintain the large cluster state in the system before single-particle measurements are performed, but there also needs to be an extra single-particle measurement system (in addition to the fusion system used to generate the cluster state) to perform the necessary single-particle measurements to receive all the qubits in the cluster state and generate the classical information necessary for the decoder to compute the syndrome graph data required to perform quantum error correction.

[0068]

[0104] Figure 3D illustrates an exemplary example of one way in which a fusion site is implemented as part of a fusion-based quantum computer architecture. In this example, qubit 1 and qubit 2 may be dual-rail coded photonic qubits, but any type of qubit is possible without departing from the scope of this disclosure. A brief introduction to dual-rail coding of photonic qubits is provided below with reference to Figures 26–29. Thus, qubit 1 and qubit 2 can be coupled to switches 621 and 623, respectively. In some embodiments, the coupling can be a waveguide, and switches 621 and 623 can be photonic switches. The various output channels of the switches can be coupled to different qubit measurement devices that perform different types of measurements. For example, single-qubit measurement device 625 (635) can perform a measurement of the state of qubit 1 (qubit 2) on an X basis, single-qubit measurement device 627 (637) can perform a measurement of the qubit on a Y basis, and single-qubit measurement device 629 (639) can perform a measurement of the qubit on a Z basis. Similarly, the two-qubit measurement devices 631 and 633 can perform different types of two-qubit measurements, for example, projected Bell measurements as referred herein in type II fusion. For example, measurement device 631 can perform XX fusion, and measurement device 633 can perform ZZ fusion or XZ fusion. Figures 31 to 36 show exemplary hardware that can be used to implement such measurement devices according to one or more embodiments. In some embodiments, the ground states of switches 621 and 623 can be hardcoded within the fusion network controller 319, or in some embodiments, the ground can be selected based on an external input, for example, an instruction provided by the fusion pattern generator 313 depending on the needs of the algorithm being executed. The layout shown in Figure 3C is intended to be illustrative only, and any number and combinations of switches and single and / or multi-particle measurement devices can be used without departing from the scope of this disclosure.

[0069]

[0105] In some embodiments, for example in linear optical embodiments, fusion can be a stochastic operation, i.e., a stochastic Bell measurement is performed, where the measurement may succeed or fail, as described below with reference to Figure 35. In some embodiments, the success rate of such an operation can be increased by using an additional quantum system in addition to the one on which the operation is acting. Embodiments that use an extra quantum system are usually called “boosted” fusion. In the example shown in Figure 3C, the fusion site performs an unboosted Type II fusion operation on incoming qubits. As those skilled in the art will see, any type of fusion operation can be applied (which may or may not be boosted) without departing from the scope of this disclosure. Additional examples of Type II fusion circuits are shown and described below for both polarization coding and dual-rail path coding. In some embodiments, the fusion network controller 319 can also provide control signals to measurement devices 625, 627, 629, 631, 633, 635, 637, 639, etc. The control signals can be used, for example, to gate measurement hardware (e.g., photon detectors) or otherwise to control the operation of the hardware. Each measuring device provides a measurement result signal (603, 605, 607, 609, etc.), which can be preprocessed at the fusion site 341 to determine the measurement result (e.g., whether the fusion was successful or not, which eigenvalues ​​were measured, how many photons were detected, etc.), or it can be passed directly to the decoder 333 for further processing.

[0070]

[0106] According to several embodiments, fault-tolerant quantum computer architectures are disclosed. In some examples, fault-tolerant linear optical quantum computers that can be fabricated on a silicon photonics platform are described. Linear optical methods to quantum computing are advantageous for several reasons, including: (i) high-coherent qubits and high-fidelity single-qubit operations can be performed using well-known quantum optical methods; (ii) silicon photonics are fabricable and provide a means to scale to a large number of qubits; (iii) all necessary operations (state preprocessing, gates, and measurements) can be performed rapidly, resulting in high gate speeds; and (iv) the dominant source of noise is optical loss, which allows for more effective error correction because the location of the error is known.

[0071]

[0107] In linear optics, two-qubit gates cannot be implemented deterministically because photons do not interact with each other. Entangled states can only be generated using operations with a success probability of less than 1. Furthermore, single-photon sources used to preprocess qubit states may not function deterministically. Overcoming this limitation introduces overhead compared to schemes with deterministic two-qubit operations. This overhead does not increase with increasing computational size. In that sense, the overhead associated with non-deterministic operation in linear optics is not as severe as the slowly increasing overhead of quantum error correction for larger computations. According to several embodiments, architectures are disclosed that can tolerate even relatively frequent failures of entangled operations, thereby significantly reducing the overhead of non-deterministic operations compared to other LOQC architectures.

[0072]

[0108] In several respects, the fact that photons do not readily interact is an advantage. This limits the possibility of so-called quantum crosstalk, where qubits can unintentionally become entangled. Such effects are a significant source of noise in many other approaches to quantum computing.

[0073]

[0109] In some embodiments, a system and method for fault-tolerant quantum computing, known as fusion-based quantum computing (FBQC), is disclosed herein. In this method, a specific, relatively small entangled state is generated, called a resource state. The computation is then performed by selecting a measurement to be performed on a pair of qubits coming from a separate, i.e., unentangled resource state. As will be described in more detail below, the measurement may be called a linear optical fusion measurement, and thus fusion-based quantum computing.

[0074]

[0110] FBQC should not be confused with measurement-based quantum computing (MBQC) techniques. MBQC techniques involve very large entangled resource states, known as cluster states, which have a number of cluster state qubits that increases as the desired number of logical qubits increases and as the desired number of gate operations in the computation increases. In MBQC, computation is performed using single-qubit measurements of the cluster states. In FBQC, the size of the resource state does not depend on either the number of logical qubits or the number of gates in the computation. As used herein, a resource state whose size does not depend on either the number of logical qubits or the number of gates in the computation is referred to as a resource state with a fixed (or constant) size. In FBQC, computation is performed by performing two-qubit measurements on qubits belonging to two separate (i.e., unentangled) resource states with a fixed size.

[0075]

[0111] In linear optical embodiments, fusion operations are probabilistic, and if they fail, this means that no result of the fusion measurement is obtained. In FBQC, quantum error correction codes can handle such missing measurement results, which are referred to herein as “erased,” and these missing measurement results can be handled using quantum error correction.

[0076]

[0112] The most efficient photonic architecture in this academic literature is based on MBQC, which uses fusion to create cluster states. The effects of fusion failures are handled using the fact that such failures result in the loss of qubits in the desired cluster state. Percolation theory results are used to ensure that if fusion failures are rare enough, the remaining cluster states have a very large connected component that can be used for MBQC. Such percolation-based architectures have significant drawbacks compared to FBQC, including the fact that the path through the remaining cluster states must be computed in real time for each logic gate, which is likely to be very difficult, and that the threshold for such schemes is actually very low.

[0077]

[0113] By using quantum error correction codes to compensate for the inevitably probabilistic linear optical entangling operation, a very high quantum error correction threshold is possible in FBQC without requiring a decoder that can also perform the computationally demanding renormalization calculations required for percolation-based methods. Percolation-based schemes require a much more complex decoder that must find a path within the percolated cluster. According to some embodiments, FBQC architectures can have a physical size (footprint) orders of magnitude smaller than percolation-based photonic architectures or alternative forms that address probabilistic linear optical operation via gate teleportation through very large ancilla states or a "repeat until successful" method.

[0078]

[0114] In some embodiments, performing FBQC involves the ability to adaptively select each measurement in response to the results of previous fusion measurements. Such adaptability can be achieved, for example, by using classical logic and appropriate switchable elements to move each qubit toward the appropriate measurement device, as previously described with reference to Figure 3D.

[0079]

[0115] FBQC combines many advantageous features. For example, preferably, every individual photon within an FBQC encounters only a small, fixed number of optical elements between the photon source and the detector, regardless of the size of the computation performed. This "constant depth" feature dramatically reduces losses compared to other architectures, as each optical element increases the probability of loss. More explicitly, photons containing resource state qubits are measured immediately in the subsequent fusion. The number of optical elements a photon passes through within the resource state generator depends on the resource state and the method used to generate it, but not on the computation performed.

[0080]

[0116] Each photon passes through a small, fixed number of optical elements (which can be, for example, 5 or less), and this number remains constant as the size of the computation increases. Therefore, the timescale for generating and detecting photons is completely decoupled from the much longer timescale required to perform non-trivial logical operations or to run the decoder. This means that the decoder does not need to be located in the same place as the rest of the computer, which is advantageous for architectures that use the cryogenic operation of quantum elements, as the decoder does not need to be located in the same place within a cryostat.

[0081]

[0117] Preferably, FBQC corresponds to a planar architecture of a computer. In such an architecture, the majority of fused measurements are between adjacent qubits within the plane of the chip. Preferably, the planar architecture makes it practical to implement non-photonic qubits on a silicon photonic chip or any other planar integrated circuit technique.

[0082]

[0118] Preferably, FBQC has enough flexibility to implement many different techniques for quantum error correction and fault-tolerant logic gates. According to some embodiments, most of the existing tools for fault-tolerant quantum computing using surface codes can be used with FBQC.

[0083]

[0119] According to several embodiments, qubit coding can be used, where the qubit is a single photon in several time bins in a given transverse mode of one of two waveguides. This is called dual-rail coding. A similarly useful variant is coding the qubit in one of two time bins traveling within the same waveguide or fiber. This is called time-bin coding.

[0084]

[0120] In dual-rail coding, each qubit has one photon, and in FBQC, all qubits are measured. Photon loss results in fewer photons detected than the expected number of photons that signal that an error has occurred. Preferably, in the FBQC methods disclosed herein, the tolerance of the surface code for such errors is much higher than that for errors that thus go unnotified.

[0085]

[0121] Another advantage of FBQC's optical implementation is its ability to generate a long delay between the resource state generator and the fusion measurement using optical fibers. This makes it possible to fuse qubits that are not simply nearest neighbors within the plane of the photonic chip.

[0086]

[0122] III. Example of FBQC Architecture According to some embodiments, FBQC can be constructed based on two primitive operations, referred herein as fusion: the generation of small, constant-size entangled resource states and projection entanglement measurement.

[0087]

[0123] FBQC can be applied across many physical systems and is particularly relevant to architectures where multi-qubit projection measurements are native operation. One or more embodiments implement FBQC in linear optical quantum computing. In the examples disclosed herein, a fault-tolerant threshold of 24% for fusion failures has been demonstrated (compared to the previously reported 14.9%).

[0088]

[0124] A. The Principle of FBQC In FBQC, the fusion network defines the configuration of fusion measurements performed on qubits of a set of resource states. The fusion network forms a computational framework from which algorithms can be implemented by modifying at least some basis values ​​of the fusion measurements. The computational output is obtained by appropriately combining the fusion measurement results. An example of a two-dimensional fusion network is shown in Figure 4A. In general, no specific structure is required within the fusion network.

[0089]

[0125] The construction of a fusion network involves two basic primitives. The first is resource state generation, which describes the generation of small entangled states. These states have a fixed size and structure, regardless of the size of the computation in which they are used for implementation. Resource states can be of any size, and the particular form of a resource state is generally not important to FBQC, but rather is a design parameter that quantum engineers can freely use considering a particular qubit architecture and noise model. In some embodiments, a resource state generation device generates copies of this resource state over a period of time referred to herein as a “clock cycle”. Resource state generators can take many physical forms, for example, they may be devices that generate entangled photonic states, or they may be material-based devices.

[0090]

[0126] The second primitive is the fusion measurement, which is a projection entanglement measurement relating to multiple qubits. In some embodiments, the fusion measurement can be performed by a fusion device having n input qubits that output n classical bits that give the measurement result. For example, a Bell measurement relating to two qubits that yields results X1X2 and Z1Z2. At least some of the fusion devices (or the generated resource states) must be reconfigurable so that the projection measurements they perform can be modified at different time steps to conform to the computational intent of FBQC, i.e., to perform quantum applications.

[0091]

[0127] The physical implementation of fusion depends on the underlying hardware. In linear optical systems, fusion can be performed natively by performing interference photon measurements that encompass different resource states, which simply corresponds to a proper configuration of beam splitters and photon detectors. More subtle implementations are also possible to improve the probability of success and robustness against hardware imperfections.

[0092]

[0128] Other approaches to quantum computing also utilize entanglement measurements performed throughout the computation. In particular, in fault-tolerant circuit diagrams, syndrome extraction can be understood as an entanglement-based batch measurement. In topological quantum computing, coupled nonionic charge projections are required to extract classical results from a system, and these coupled nonionic charge projections can be used as a foundation for achieving universal quantum computing. Redundancy of fused measurement results can be used to naturally accommodate the constant density of syndrome extraction required to alleviate entropy accumulation.

[0093]

[0129] B. Architecture FBQC provides a natural framework for studying fault tolerance by considering resource state and fusion primitives, but its advantages also lead to a significant simplification of physical architecture requirements. In addition, a third component, a fusion network router that enables the first two to work together, can also be explicitly identified for the generation and fusion of resource states by appropriately routing qubits from resource states to fusion measurements. Integrated waveguides and optical fibers enable direct and low-loss routing of photonic qubits over very long distances, but other material-based methods require coherent optical-matter coupling, which has only been demonstrated with relatively low fidelity; therefore, the fusion network router offers the greatest advantage in realizing linear optics.

[0094]

[0130] A given fusion network has many possible architectural implementations, for example, for a 3D fusion network, and it can choose to create all resource states simultaneously, or alternatively, reuse resource state generators to create a new copy of the state in each clock cycle, creating one 2D layer at a time. This architectural design is taken up by a fusion network router that guides qubits created at different spatial and temporal locations (i.e., from different resource state generators and time bins) to the corresponding fusion locations. Thus, the fusion network router includes both spatial and temporal routing in the form of delay lines. Figure 4B shows a schematic example of an FBQC architecture that creates a 2D fusion network from a 1D array of resource state generators.

[0095]

[0131] In certain fault-tolerant fused networks, the fused network routers to implement a fixed routing configuration. Fixed routing means that qubits generated from a given resource state generator are always routed to the same location. This design feature is particularly attractive from a hardware perspective and has many practical implications. Specifically, it minimizes the need for potentially error-prone switching and reduces the burden of classical control.

[0096]

[0132] Another important feature of the FBQC architecture, and what distinguishes them from other methods, is the separation of timescales for classical control. As shown in Figure 4B, feedforward control can be performed at the logic level to process and decode the measurement results, which then influences future logic operations. However, this timescale can be orders of magnitude longer than the clock cycle of resource state generation and fusion, and classical computation or feedback is not required at this shorter timescale. In other words, the physical qubits do not need to wait in memory while computations are being performed to determine how they should be measured.

[0097]

[0133] IV. Fusion In FBQC, the initial quantum resources are small, fixed-size, entangled resource states. The large-scale quantum correlations required for universal computing are generated when measurements are performed on qubits from different resource states. For this to generate long-range entanglement, at least a portion of the measurement results must be entangled, i.e., the projector must be on a subspace containing at least one entangled state.

[0098]

[0134] In general, the measurement can be any positive operator evaluation measure (POVM), but for the purpose of achieving fault tolerance, it is useful to consider measurements where all results are projections onto stabilizer states. This makes it easier to use existing stabilizer fault tolerance methods. In the examples in this paper, we focus in particular on the case of a two-qubit measurement which is a Bell state projection, and we will refer to this as Bell fusion. Bell fusion measures input qubits in stabilizer basis X1X2, Z1Z2.

[0099]

[0135] Generally, we focus on fused networks where most of the fused measurements required to perform quantum error correction are identical Bell measurements. However, to implement logic gates, some of the fused measurements need to differ from others. There are many variations on how this can be achieved, either by using two-qubit measurements on a modified stabilizer basis or by including single-qubit measurements. This will be discussed in more detail below.

[0100]

[0136] C. Fusion in linear optics In linear optical quantum computing (LOQC), fusion of photonic qubit pairs is straightforward but does not deterministically generate entanglement. This nondeterminism means that the desired measurement results may not always be obtained, and preferably, one or more embodiments of the architecture for LOQC find a way to avoid this missing information. In the FBQC scheme, these fusion failures are directly corrected by quantum error correction.

[0101]

[0137] The example studied here specifically considers a “dual-rail” qubit consisting of a single photon of two photonic modes. The photon of the first mode represents logic |0>, and the photon of the other mode represents logic |1>. This qubit coding is attractive because it is lossy and the qubit is taken out of the computational subspace and therefore known. Bell fusion on dual-rail qubits can be performed using a linear optical circuit in which all four modes of the two qubits are measured. This is often called a type II fusion. The fusion is 1-p fail The process "succeeds," measuring the input qubits of the bell stabilizer basis X1X2 and Z1Z2 as intended. The fusion is probability p fail If it "fails," then separate single-qubit measurements Z1I2, I1Z2 are performed. If there is a possibility of photon loss or other imperfections, there is a third possible outcome: fusion "erasure." In this case, the intended stabilizer result is not measured. Figure 5 shows different possible measurement results for linear optical fusion in two qubits.

[0102]

[0138] Figure 5 shows the results of a linear optical Bell fusion. Fusion on qubits from two cluster states is shown, with the intended results X1X2 and Z1Z2. In the presence of photon loss, there are three possible outcomes: a successful fusion where both measurement results are obtained, a failed fusion where only result Z1Z2 is obtained, and a fusion annihilation where no measurement result is obtained. Fusion failure is inherent to linear optics and can occur even when all operations are ideal. Fusion annihilation most commonly occurs only due to a system error, when one or more photons entering the fusion measurement are lost.

[0103]

[0139] A failure in linear optical fusion is a benign error, rather than annihilation, since it is known, and does not result in a mixed state, as a pure stabilizer measurement is still obtained. One of the two desired results, Z1Z2, can be obtained by multiplying two single-qubit measurements. Thus, a fusion failure can be treated as a Bell measurement where the X1X2 measurement result is annihilated.

[0104]

[0140] The simplest way to implement Type II fusion involves only two beam splitters and four detectors, with a failure probability of p fail It has a success rate of 50%. By using additional Bell pairs, the fusion can be "boosted" to reduce the failure probability to 25%, and by using more auxiliary photons, the fusion success rate can be further boosted. Increased resistance to photon loss and physical fusion failure can be achieved by performing the fusion on the encoded qubit. This method is used in the following example, where the physical qubit is encoded using a (2,2)Shor code, and the encoded fusion is performed by performing the physical fusion laterally. Below, we explain how erasure can be suppressed in encoded fusion and calculate the probability of erasure of the measured value from encoded fusion in the presence of photon loss and fusion failure.

[0105]

[0141] To implement the logic, the measurement basis of linear optical fusion can be easily modified by placing a single-qubit rotation before their inputs. These single-qubit gates can be implemented with high precision using beam splitters and phase shifters, which are easy to implement on an integrated photonic chip. A small switching network before fusion allows for reconfigurability between different measurements, which will be discussed further below.

[0106]

[0142] V. Resource Status In FBQC, small entangled states that affect a computation are called resource states. Importantly, their size is independent of the computation being performed or the sign distance used. This allows them to be generated by a certain number of consecutive operations. As a result, resource state errors are limited, which is important for fault tolerance.

[0107]

[0143] Similar to fusion, we focus on the resource states of a qubit stabilizer. Such states can be described by a graph G using a graph state representation, down to local Clifford operations, where the described quantum state |G> is obtained by placing a qubit at each vertex into |+> and the corresponding vertices in the graph perform controlled Z gates between adjacent qubits. Similarly, an n stabilizer generator of a graph state with vertices labeled from 1 to n is:

number

[0108]

[0144] Figure 6 shows an example of a resource state represented as a graph state according to one embodiment. Figure 6A shows an example of a resource state in the form of a 6-ring graph state. With qubits labeled as shown in the figure, the stabilizers of the resource state are Z6X1Z2, Z1X2Z3, Z2X3Z4, Z3X4Z5, Z4X5Z6 and Z5X6Z1. Figure 6B shows that the qubits in the resource state can be replaced by a (2,2)Shor encoded resource state having the illustrated transformation. Both qubits 1 and 2 have neighboring qubits drawn as dotted circles, the same as the unencoded qubit on the left. Qubits with H inside have Hadamard applied to them with respect to their graph state representation. Figure 6C shows the resource state of Figure 6A with all qubits encoded with (2,2)Shor codes.

[0109]

[0145] The stabilizers for this resource state are Z6X1Z2, Z1X2Z3, Z2X3Z4, Z3X4Z5, Z4X5Z6, and Z5X6Z1. The resource state can be encoded into (2,2)Shor codes by following the transformation shown in Figure 6B. Replacing all the qubits in the 6 ring with (2,2)Shor encoded qubits yields the resource state shown in Figure 6C.

[0110]

[0146] The operations used to create resource states depend on the physical platform used for this process, which may differ from the physical platform used to implement the fusion network, insofar as the generated resource state qubits are compatible with the fusion network. For example, with solid-state qubits, resource states can be generated using unitary entanglement gates or dissipatively. When using linear optics, resource state generation is achieved by performing a series of projection measurements, such as fusions, on smaller entangled states, such as Bell states and 3-GHz states, sometimes called seed states. Methods for generating seed states are fully covered. Since projection entanglement measurements in linear optics are probabilistically successful, it is often advantageous to use switching networks between fusions to increase the success rate of the protocol, as discussed above. These networks are used to attempt the probabilistic operations multiple times and select only the successful cases. In this sense, multiplexing is used to effectively approximate the selected state when entangling the results of fusions. Since the size and number of probabilistic operations required to generate resource states are fixed, the overhead from repeating the probabilistic operations is also fixed. There are many options for implementing such switching networks, and the latest scheme can be found depending on the required efficiency and available devices. It is worth noting that while the resource state must be a qubit state, i.e., a state with multiple parts of entanglement between clearly defined qubits, states acquired in the intermediate stages that generate the resource state do not need to adhere to this restriction.

[0111]

[0147] Since the noise profile of a resource state depends on the generation protocol used, determining the most appropriate resource state is part of designing an FBQC scheme for a practical hardware implementation. For a given target resource state, there are a vast number of possible preprocessing protocols, each resulting in a different noise profile. However, the fixed size of the resource state means that any generation protocol requires a finite number of operations, and therefore the noise accumulated in any of the state generation is limited. Furthermore, any error correlations arising from independent state generation are local to that state, limiting the spread of errors in the fused network, as explained below.

[0112]

[0148] VI. Converged Networks A fused network (FN) specifies the resource states used in the FBQC protocol and how they are connected by fusion. After fusion measurements are performed, two types of information remain: classical information from the measurement results and several quantum correlations corresponding to (potentially) unmeasured qubits. These measurement results include correlations that are the "results" of the fused network, providing both a computational output or, in the case of a fault-tolerant fused network, a parity check that can be used for error correction. This section describes how to construct a fused network and how to analyze them to identify the quantum and classical correlations that exist after fusion measurements have been performed. In particular, we will focus on stabilizer fused networks where the resource states are stabilizer states and the fused measurements are stabilizer projections. This allows us to utilize existing fault-tolerant tools.

[0113]

[0149] The stabilizer fusion network can be characterized by two Pauli subgroups: (1) a stabilizer group R that describes the ideal resource state and (2) a fusion group F that is a Pauli subgroup that defines the fusion measurements, where F ∋ -1. Assuming complete fusion, by implementing the fusion network, the eigenvalues of all the operators in F are learned. Since the individual measurement results of the fusion operators are random, -1 is included. The fusion stabilizers that should have a result of "+1" according to the definition correspond to the consistently signed elements of F. After the fusion measurements are made, the remaining system can be described by the remaining stabilizer group. S := Z R (F)

[0114]

[0150] This is the centralizer of F in R. Since not all of the stabilizers of F and the stabilizers of R commute with each other, after the fusion measurements are made, only some subgroups of the original stabilizers, i.e., the remaining stabilizers, remain. These remaining stabilizers include both the "already measured" qubits (the remaining information is purely classical) and the qubits that still have quantum correlations but have not yet been measured. Any remaining qubits are the output stabilizer group S, which is a restriction of S to the remaining qubits up to a sign that can be determined from a particular fusion result. out That is, the signs of the stabilizers in S are calculated such that multiplying an element of the fusion group F that has a sign that matches the obtained measurement results produces an element of S. out

[0115]

[0151] In the example of the fault-tolerant fusion network disclosed herein, all qubits in the network are measured and there are no remaining qubits. The computation in FBQC uses the correlations between the fusion measurements that arise from the structure of the fusion network.

[0116]

[0152] (a) Three resource states: An example of a fused network having two copies of a 2-qubit graph state and a 3-qubit linear graph state. There are two fused states, shown by the orange lines, both of which measure the operator <XZ,ZX>. Specifically, the resource state consisting of qubits {1,2,3} is stabilized by <Z1X2Z3,X1Z2I3,I1Z2X3> and similarly with respect to {6,7,8}. Qubits {4,5} are stabilized by <X4Z5,Z4X5>. If all measurements in the fused network are successful and return a +1 eigenvalue, the unmeasured qubits {1,2,7,8} are stabilized by <X1Z2,Z1X2Z7,Z2X7Z8,Z7X8>, which correspond to the 4-linear graph state shown in (b).

[0117]

[0153] A simple example of a fusion network is shown in Figure 7A, which leads from the state to the result-dependent stabilizer sign in Figure 7B. The resource state group R is generated by the union of stabilizers of different resource states, which can be inferred from their graph state representations. R = <X1Z2, Z1X2Z3, X4Z5, Z4X5, Z2X3, X6Z7, Z6X7Z8, Z7X8>. The fusion group is generated by the union of all fusion measurement operators, i.e., F = <X3Z4, Z3X4, X5Z6, Z5X6, -1>. The classical information generated by the fusion network is obtained from the measurement results of F. Furthermore, the fusion network is represented by the stabilizer S shown by the graph state in Figure 7B. out The quantum information regarding the unmeasured qubits having the values ​​=〈±X1Z2,±Z1X2Z7,±Z2X7Z8,±Z7X8〉 is preserved. out The sign depends on the fusion measurement results.

[0118]

[0154] Some aspects of the FBQC architecture are not covered by the fusion network. In particular, the fusion network does not capture the temporal order of fusion, physical qubit routing, or classical processing requirements. The fusion network does not specify the order of fusion and can be executed in the order most appropriate for the underlying hardware. Furthermore, the resource states involved in the fusion network do not all need to exist simultaneously, and their generation may be staggered, as long as fusion measurements can be performed for all necessary qubit pairs.

[0119]

[0155] The following explains how redundancy can be added to a fused network for fault tolerance.

[0120]

[0156] VII. Fault-tolerant fusion networks A fusion network can be constructed to be fault-tolerant, as long as the probability of errors is sufficiently low, so as to compensate for errors in resource states or noisy fusion circuits that result in fusion measurement results that deviate from their ideal values. This section describes fault tolerance in fusion networks. Fault-tolerant fusion networks (FTFNs) can be constructed in a manner inspired by circuit-based quantum error correction, or based on fault-tolerant cluster states. While this approach can serve as a useful initial guide, direct conversion often results in inefficient schemes; better schemes can be found by working more directly with the fusion network diagram, as demonstrated in the examples herein.

[0121]

[0157] A. Stabilizer format for FTFNS When modeling physical errors from both resource state preprocessing and fusion measurements, they are interpreted as occurring in the space before all resource states are preprocessed and fusion is performed. In this diagram, the key to fault tolerance is the redundancy between the Pauli operators F measured during fusion and the stabilizers of the resource states R. This redundancy is reflected in the presence of check operator groups. C:=R∩F

[0122]

[0158] In other words, check group C corresponds to a subset of stabilizer R on the resource state made available by fused measurement group F. If there are no errors, the fused result should be compatible with all operators in C that have positive eigenvalues; that is, the classical fused result forms a classical linear binary code, which makes correction possible. Error correction processing is described below. The group of undetectable errors is defined by the centralizer of check group C for the entire Pauli group. U:=Z(C)

[0123]

[0159] As the name suggests, this subgroup of Pauli operators leaves no trace on the result of the check operator when applied to pre-fusion qubits in other ideal processes. However, not all undetectable errors are problematic for computation, as some do not affect the final correlation of the subject. For example, any element of R or F may have no harmful effect, as the elements of R may leave the resource state invariant and the elements of F may leave the fusion invariant, which can be absorbed into ideal state pretreatment or fusion measurement, respectively. More generally, a small group of undetectable errors may have no harmful effect. T:=Z(S),

[0124]

[0160] This includes <R,F> by definition. Therefore, undetectable errors can be classified by the elements of the quotient U / T, where S is the residual stabilizer group defined below. Two errors are considered distinct if they result in different syndromes or have different logical effects. In contrast, if they differ only by the elements of T, they are equivalent in all relevant ways. Thus, equivalent errors correspond to the equivalent class P / T.

[0125]

[0161] Figures 8A–8C show examples of fused networks along with explicit definitions of these groups. All fusions measure input qubits at base XX, ZZ. Output stabilizer group S out =〈±X1X8,±Z1Z8〉, that is, the fusion network generates a Bell pair when all measurements are successful. Figure 8A shows an example of a fusion network where resource state group R, fusion group F, and check group C may be explicitly defined in Figure 8B. Figure 8B shows that the set of resource state group generators is the union of stabilizers of different resource states, which can be inferred from their graph state representations. The green fusion measures the input qubits in basis XX,ZZ. The fusion group is generated by all fusion measurements and -1. The generators from different resource states in R and different fusions in F are sorted by column. Output stabilizer group S out =〈±X1X 16 ,±Z1Z 16 > In other words, the fusion network generates a Bell pair when all measurements are successful. out The sign depends on the specific measurement results. Figure 8C shows a syndrome graph from a fused network (a) with measurements corresponding to all labeled edges. Multiplying the adjacent measurements by the check yields the same generator of C as in Table (b) S = 〈X1Z4Z 13 Z 16 ,Z1Z2Z3X4Z5X6Z7X8X9Z 10 X 11 Z 12 X 13 Z14 Z 15 Z 16 ,C〉

[0126]

[0162] Since F⊆T, there is no need to distinguish different errors that are equivalent down to the elements of F. We choose to represent the decoding problem with respect to the elements of P / F (i.e., the complete Pauli group commercialized by the fusion group). Distinct elements of P / F can correspond to equivalent errors (according to the fully reduced equivalence class of P / T), but this reduction has the advantage of preserving a large amount of locality structure in the error model. In particular, a single-qubit Pauli error on a resource state is interpreted as a measurement error on the corresponding generator of F. If F consists of Bell fusion measurements, the quotient P / F identifies a pair of single-qubit Paulis in P whenever the elements of F are produced together. Thus, we can choose to represent the decoding problem with respect to P / F which directly corresponds to specifying which fusion result has been inverted. For example, in the example in Figure 4, single-qubit errors X4 and X 13 X4X is an element of F. 13 Since it is multiplied by this, it is an equivalent error. Up to this equivalence, the error is determined by which generator (fusion result) of F is inverted, in this case Z4Z. 13 It can be characterized by the following.

[0127]

[0163] The most preferable type of error is a trivial error corresponding to an element of T. However, they may have non-trivial indications in P or with respect to the fused result being reversed. For example, measurement results X2X5 and X 10 X 14 If both are reversed (i.e., in some cases error Z2Z) 10 (Due to this), neither check is affected, so Z2Z 10 ∈U is an undetectable error. Error Z2Z 10 It also commutates with the remaining generator of S, which also results in a slight error (i.e., Z2Z). 10 ∈T) and this means that it does not affect the predicted sine of the output stabilizer. X2X5 and X 10 X14 A physical error that causes the inversion is Z2Z 10 It is not important whether it was X²X⁵ or another operator in T. In fact, X²X⁵ and X 10 X 14 Any combination of the results is equally possible in the idealized setting, extract the check operator, and output stabilizer S out When calculating the sine of a given number, only the combined parity (XOR) of those numbers is used.

[0128]

[0164] A worse type of error is a non-trivial, undetectable error. This is Z4 or Z, which belong to U but not to T. 13 X4X can be generated by single-qubit errors such as the following. 13 This is the case when the result flips. That is, these errors commutate with the check operator of C and therefore go undetected, but reverse with one of the additional generators of S. This leads to an incorrect prediction of the sine on ±Z1Z6 on the output stabilizer.

[0129]

[0165] The most interesting case for fault-tolerant fusion networks is the case of detectable errors. Z4Z 13 Measurement errors in this area result in inconsistencies in both check generators in Figure 4B, thus making the errors detectable (i.e.,

number

[0130]

[0166] For fault tolerance, we are interested in fused networks where the weight of non-trivial, undetectable errors, also known as the sign distance, increases with the network size. Several examples of such networks are described below. In the large periodic networks discussed later, it is inconvenient to explicitly write the complete resource state group R and fusion group F, as done in Figure 4. Instead, we specify only the stabilizer measured for a single resource state and a single fusion. Resource state and fusion group generators can be obtained by repeating the same stabilizer for different sets of qubits. Instead of explicitly writing check groups, it is more convenient to represent checks graphically using the “syndrome graph” described later.

[0131]

[0167] B. Local FTFN and Topological FTFN The concept of a local FTFN, which is completely similar to other similar configurations in error correction, can be introduced. That is, the family of stabilizer FTFNs is

[0132]

[0168] It is local if it has a local check operator generator (i.e., each generator contains a limited number of fusions, and each fusion involves a limited number of generators), and there exists a family of fusion networks such that for any integer d, a non-trivial, undetectable error supports at least d qubits.

[0133]

[0169] Next, the existence of an error threshold can be demonstrated by applying the usual combination arguments. For example, if an error model is adopted in which fusions may be erased or randomly flipped, for any given local FTFN, there exists a subthreshold region in the plane defined by the erase rate and flip rate such that the FT can be achieved. That is, as long as the combination of the erase rate and flip rate is below the threshold, any desired logical error rate can be achieved by constructing a sufficiently large fusion network.

[0134]

[0170] An important class of local FTFNs is the topological fusion network. Using the schematic as a reference, most topological fusion networks can be considered as mimicking the FT error correction process of topological codes. As a result, in the case of 2D topological codes, a 3D topological fusion network is obtained in which the elements of the surviving stabilizer group S take the form of membranes (word lines of string operators) and the undetectable errors U take the form of closed strings (word lines of phase charges). While there is a described equivalence between these 3D topological fusion networks and 2D topological codes, 3D topological fusion networks are not constrained to represent leaf codes and can support a more general framework of measurement-based fault tolerance described. An example of a topological FN based on a toric code is described below.

[0135]

[0171] C. Composite Diagram The redundancy of classical codes associated with fault-tolerant fusion networks is often adequately explained by syndrome graph representations, facilitating the application of existing decoders, such as minimum-weight matching and union-find decoders, to the FBQC framework. Another advantage of syndrome graph images is that they provide a graphical way to design fault-tolerant schemes with higher thresholds. However, it should be noted that not all schemes can be naturally represented by syndrome graph structures.

[0136]

[0172] A syndrome graph is a graph representation of a classical linear code using a multigraph, where vertices correspond to check generators and edges correspond to mutable nodes. We use them so that each vertex represents a check operator in C, and each edge represents a generator (or more precisely, its error) in a fusion group F. An edge is connected to a vertex if the corresponding generator in F is used in the factorization of the corresponding check operator in C. Note that given a set of independent generators in a fusion group F, each element of check group C has its own unique factorization with respect to it. Furthermore, note that it depends on a fault-tolerant fusion network and the choice of generators for C to ensure that each edge is connected to at most two vertices (i.e., each fusion generator is used by at most two check generators). What makes this possible in topological FBQC is the manifestation of the fact that error chains leave non-trivial syndromes only at their endpoints.

[0137]

[0173] The parity of a check operator is evaluated by taking the combined parity of all the measurement results that make up the check. Given a set of fused measurement results, each parity check has an associated parity value of either +1 or -1. The composition of all these parity results is called a syndrome. If a fused result is inverted, checks incident on that edge in the graph have their parity values ​​inverted. If a fused result is erased or missing, two checks incident on an edge in the graph can be multiplied / combined into a single check operator. A syndrome graph can have "unconnected edges" where an edge connects to only one check vertex, in which case the check / vertex is removed if the fused result is erased or missing. In some cases, multiple fused measurement results can also exist between two check nodes if multiple fused measurement results contribute to the same pair of syndromes.

[0138]

[0174] Figure 4C shows how an example of a fused network can be represented as a syndrome graph. In this simple example, there are two check operators, each with four incident edges. There is one edge shared by both check operators that connects two vertices. The other edges are "unconnected edges" connected to only a single check node. In this small example, not all fusion results are contained within a check operator, but in the topological fault-tolerant fused network presented in the next section, all fusion results will be part of at least one check operator. Specifically, consider topological fused networks based on surface codes. As with any surface code construction, these networks can be represented by locally cut primal syndrome graphs and dual syndrome graphs. In the bulk, every Bell fusion contributes to two measurements, which correspond to one edge in the primal and one in the dual syndrome graph, respectively.

[0139]

[0175] VIII. Exemplary Fault-Tolerant Fusion Networks This section describes two explicit examples of fault-tolerant fused networks that implement surface code error correction. These examples provide a brief explanation of how fault tolerance can be achieved within the FBQC framework. They are selected as useful educational examples and are not optimal FBQC architectures. Nevertheless, even using these examples, the inventors demonstrate significant performance improvements.

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[0176] A. "4-star" converged network According to several embodiments, a "4-star" fusion network is shown in Figure 9. The resource state is a 4-qubit Greenberger-Horne-Zeilinger (GHZ) state with stabilizers Z1Z2Z3Z4, X1X2I3I4, I1X2X3I4 and I1I2X3X4. For clarity in the graph, this resource state is represented as a 5-qubit star graph state with the central qubit blacked out (Figure 9A), because this state is obtained when measuring the central qubit of a 5-star graph state in an X basis with a "+1" result. There is no need to preprocess the 5-qubit physical resource state, and the 4-GHZ state can be created directly. For example, the 4-GHZ state can be preprocessed from a single photon using a linear optical system and the circuit described. The four purple circles correspond to the qubits of the resource state and are input to the fusion in the network. The fusion network can be constructed from a cubic unit cell, as shown in Figures 9A and 9D, in which case resource states are located on all faces and edges of the unit cell. Resources are aligned parallel to the faces or perpendicular to the edges. As shown in Figure 9B, fusion measurements are performed on pairs of qubits from resource states centered on a unit cell face and qubits from resource states centered on an adjacent edge (if the central qubit of a resource state is not measured, the fusion results in a cluster state used in the MBQC implementation of the surface code). Each fusion attempts to measure stabilizer operators X1Z2 and Z1X2, as shown in Figure 9C.

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[0177] The fused network yields the syndrome graph shown in Figure 9E, i.e., a cubic lattice, where all edges are 4-directional multi-edges corresponding to four measurement results. There are a total of 24 fused measurements, which are combined to evaluate each check operator. The fused network is symmetric under translation by half the lattice constant in all three dimensions. This means that the primal syndrome graph and the dual syndrome graph are identical.

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[0178] Since syndrome graphs are used across hardware implementations of surface codes, understanding the correspondence between FBQC and circuit-based surface code implementations is a useful tool. In the circuit model, the spatial edges of the 3D syndrome graph correspond to physical qubit errors, and the temporal edges correspond to measurement errors. In FBQC, both temporal and spatial edges correspond to fused measurement results, and in this model there is no distinction between physical and measurement errors. Another point of comparison is the interpretation of primal and dual syndrome graphs. In the circuit model, the primal syndrome graph captures Pauli-X errors and measurement errors with Z-type parity checks, while the dual syndrome graph captures Pauli-Z errors and measurement errors with X-type checks. In this FBQC example, each two-qubit fusion contributes one measurement result to the primal graph and the other to the dual graph. One way to see this is that the two fused measurement results behave like the Pauli-X and Pauli-Z portions of the error channel on the physical qubit in the circuit model.

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[0179] B. "6 Rings" Convergence Network Our second example, the 6-ring fusion network shown in Figure 10, improves upon the 4-star network. It requires fewer resource states and fusion measurements to implement the same distance codes, and as we will see in the next section, this provides a significantly improved threshold.

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[0180] In this fusion network, resource states are graph states in the form of a 6-qubit ring. The fusion network has cubic unit cells with two resource states per unit cell, as shown in Figures 10A and 10D. Fusion measurements connect pairs of qubits at each face and edge, as shown by the orange lines in Figure 10B. Each fusion attempts to measure the stabilizer operators X1X2 and Z1Z2 on the input qubits. Figure 10 shows multiple unit cells, and it can be seen that the resource states form layers in a plane perpendicular to the (1,1,1) direction. The following sections provide formal definitions of 4-star and 6-ring fusion networks.

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[0181] The syndrome graph of this fused network is shown in Figure 10E, and is a cubic lattice with added diagonal edges. Every check vertex has 12 incident measurements, and half of the 24 measurements contribute to each check operator in the 4-star network. The 6-ring network has the same symmetry under translation in all three dimensions by half of the lattice constant. Therefore, as with the 4-star network, the graphs of the primal and dual syndromes are identical.

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[0182] The diagonal edges appearing in the syndrome graph here are a well-known feature in circuit-based surface coding, interpreted as so-called "hook" errors, where a single error event spreads to adjacent qubits in the stabilizer measurement circuit. The causes of this type of correlation error differ considerably in fused network settings, but their appearance in the syndrome graph is the same.

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[0183] C. Performance comparison The inventors study the performance of these two fused networks by simulating their behavior under both Pauli error and erasure models. Consider the two error models.

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[0184] An error model that can erase all fused measurements and reverse them with some probability.

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[0185] A linear optical error model in which every fusion has a probability of failure and a probability of all photons in the resource being lost.

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[0186] The inventors perform Monte Carlo simulations of each fused network with L×L×L unit cells and periodic boundary conditions in all three dimensions. Based on the above model, error samples are plotted, and decoding is performed using the simplest version of a union-find decoder to count instances of logical errors. This simulation is repeated over a range of values ​​for the extinction probability and Pauli error rate, as well as the system size L, to construct threshold surfaces for the two error parameters. Primal and dual syndrome graphs are decoded separately.

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[0187] D. Pharmacological Error Model Two error parameters, namely, the fusion elimination probability p. erasure and measurement error probability p error Threshold curves in star (blue line) and 6-ring (orange line) fusion networks with linear optical failure. A union finder decoder is used here. The green star indicates the operating point with linear optical failure erasure when the resource state qubits are encoded with (2,2)Shor codes and all fusions are boosted to a 75% success probability with a randomly selected physical failure basis. The effective erasure probability of encoded fusions with linear optical failure and loss is calculated as follows:

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[0188] Every fusion within a fusion network yields two measurement results called fusion measurements. The phenomenological error model is an independent and identical error model on the fusion measurements, and all measurements within the fusion network have probability p erasure It is eliminated by probability p error This is then inverted. This makes it possible to capture single-qubit Pauli errors and erasures originating from resource state generation, as well as those originating from the fusion device itself.

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[0189] Compared to previous studies of fault-tolerant MBQC that investigated the extinction and error thresholds of single-qubit measurements on lattices already entangled in long-range distances, this model incorporates the error of the batch measurements used to generate long-range entanglement starting from small resource states. Thus, what we call the phenomenological error model is closer to a circuit-level error model in which individual resource states and fused measurements play the role of elementary gates.

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[0190] Figure 5 shows the threshold curves (obtained by a union finder decoder) for fused networks of 4-star (blue line) and 6-ring (orange line) networks with this error model. The extinction probability p per measurement is shown. erasure and error probability p error If the combination falls below the threshold curve, the error lies within the correctable region of the corresponding fused network. Within the correctable region, the probability of non-trivial, undetectable errors, also known as logical errors, is suppressed exponentially with respect to the network size.

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[0191] The modifiable region of a 4-star network is included in the modifiable region of a 6-ring network, that is, it can be modified by a 4-star network (p erasure , p error Any value of ) can also be modified by a 6-ring network. Limit p of a 4-star network erasure The threshold is 6.9%, but the critical degradation threshold for the 6-ring network is 11.9%. error The threshold (0.94%) is also higher than the threshold for the 4-star network (0.65%). Therefore, the 6-ring network can be said to be more fault-tolerant than the 4-star network.

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[0192] A common observed trend is that fusion errors in fault-tolerant fusion networks can be improved by relying on larger resource states, as the phenomenological threshold is high. This is because the phenomenological error model only captures errors in the fusion graph portion and not internal errors in the resource states. Generally, the complexity of the resource state generator increases with the size and degree of entanglement of the generated states. The trade-off between keeping this complexity as low as possible and raising the phenomenological threshold for fusion errors is one of the key optimization goals in designing fault-tolerant FBQC architectures.

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[0193] E. Linear Optical Error Model Photon loss probability p loss and the probability of fusion failure p fail The threshold curves for 4-star (blue) and 6-ring (orange) fused networks under a linear optical error model with the following properties are for randomly selected coding and failure bases. The green curve corresponds to a 6-ring fused network with qubits coded with (2,2)Shor codes. The error model used to evaluate these curves is described below.

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[0194] Here, we examine the performance of these fusion networks under an error model motivated by linear optics. All fusions are linear optics "Type II" fusions with four photons, namely two photons from the qubit being measured and two photons from the Bell pair used to boost the probability of fusion success. Each of these four photons, including the two boosting photons, is probabilistically...

number

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[0195] To further reduce the extinction probability, consider the case where all qubits in the resource state are replaced with qubits encoded in (2,2)Shor codes, as shown in Figure 6. This replacement replaces the fusion between (unencoded) qubits in the resource state with coded fusion between coded qubits in the resource state, which consists of pairwise fusion between the physical qubits that make up the coded qubits. The fusion measurements in the network are replaced with coded fusion measurements.

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[0196] The (2,2)Shor code refers to a 4-qubit [[4,1,2]] quantum code that can be obtained by concatenating repeating codes of X and Z observables. Depending on the order of concatenation, the resulting code space is described by a code stabilizer (<XXXX,ZZII,IIZZ> or <ZZZZ,XXII,IIXX>). For simplicity, we assume that this selection is made uniformly randomly for each encoded fusion. The encoded fusion on 2 qubits A and qubit B is an encoded Bell basis

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Number

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[0197] The encoding we are modeling separately has three levels. That is, the lowest level is the encoding inherent in linear optics that represents each physical qubit as a dual-rail photon. Then, local encodings such as the (2,2) Shor code can be used for the resource state qubits to achieve encoded fusion that is less affected by the failures and losses of linear optical fusion. Finally, a fusion network such as a 6-ring network consisting of many resource states and fusions defines topologically protected logical qubits.

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[0198] The green star in Figure 11 indicates the magnitude of the encoded erasure probability p fail = 0.043 when it is only due to the fusion failure probability p enc = 25%, which is the value achieved by boosting with Bell pairs. This number can be obtained by combining the expressions of p0 and p enc in the previous paragraph of this subsection and is explained in detail below.

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[0199] (2,2) In Shor coding, erasure from fusion failures falls within the correctable region for both 4-star and 6-ring networks. In the case of a 6-ring network, the gap between this baseline operating point and threshold curve is significantly larger than in a 4-star network. The baseline erasure rate is less than half of the erasure periphery, leaving room for other errors such as photon loss.

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[0200] Figure 6 numerically examines the loss tolerance when fusion failures are present. The blue and orange lines represent the threshold curves for 4-star and 6-ring fusion networks without qubit coding, respectively. However, the failure thresholds for these networks are less than 25%. With (2,2)Shor coding, the 6-ring fusion network provides a significantly larger critical failure threshold of 43% and a critical loss threshold of 5.9%. With a 25% failure probability achieved by fusion boosted with Bell pairs, we have a loss tolerance of 2.7% per photon. In other words, by boosting fusion with Bell pairs, the fusion network can be brought into a reproducible region even when the probability of at least one photon being lost in the fusion is 10.37%.

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[0201] IX. Quantum Computation Using Fault-Tolerant Fusion Networks The above described how to create a fault-tolerant bulk that functions as a fabric for topological quantum computing. Creating the bulk is the most important component of the architecture, as it determines the error correction threshold. However, additional functionality is required to implement fault-tolerant computing. Here, we turn our attention to the question of how this bulk can be used to implement fault-tolerant logic, as well as the implications for classical processing and the physical architecture.

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[0202] A. Logic Gates To perform fault-tolerant logic, the systems and methods disclosed herein enable the creation of topological features in addition to bulk. There are different approaches that can be used to create fault-tolerant Clifford gate sets. Using boundaries, punctures can be created that can be braided and gated. Using boundaries, patches can be created that can be lattice-surgered. Alternatively, logic qubits can be encoded in defects and twists. All of these techniques for logic are compatible with FBQC. Such topological features can be created by modifying fusion measurements at specific locations or by adding single-qubit measurements in an appropriate configuration. Herein, an example of a method for creating two types of boundaries is given to facilitate the encoding and manipulation of logic qubits in punctures or patches. In some embodiments, these two boundary types correspond to rough and smooth boundaries in surface coding diagrams, but in FBQC, it is more natural to refer to them as primal boundaries and dual boundaries, depending on whether they can match excitations in primal / dual syndrome graphs, respectively. The primal boundary corresponds to the coarse boundary of the primal syndrome graph and the smooth boundary of the dual syndrome graph, as shown in Figure 22C, for example.

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[0203] Figure 13 shows an example of how primal and dual boundaries can be generated by measuring specific qubits in the Z basis. Figure 13A shows the generation of a boundary, where the layer of qubits at the boundary of the unit cell is measured in the Z basis or simply not generated. Figure 13B shows a similar protocol for creating a smooth boundary. It is particularly simple when the boundary is parallel to a plane defined by a pair of unit cell vectors. In such a boundary, the only difference between the primal and dual cases is the displacement of half the unit vectors in the vertical direction.

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[0204] The effect of this measurement pattern is to terminate the bulk and create boundaries that can be used as features for encoding and manipulating logic qubits. Figure 13D shows an example of how these boundaries are macroscopically assembled to preprocess the state |0> (or |1>) in patch-encoded logic qubits in a fault-tolerant manner.

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[0205] B. Pulse frame tracking In FBQC, logical states have only direct physical counterparts up to the Pauli correction, which are tracked in classical logic via so-called Pauli frames, for example, within the classical computing system 307 in Figure 3A. The use of Pauli frames is inherently unavoidable due to the randomness introduced by telemetry performed by Bell measurements. For example, at the logical level, the same component that preprocesses the |0> state in Figure 13 also represents the preprocessing of the |1>=X|0> state. In general, this means that for some tracked Pauli correction operator P, a state |ψ> can be physically represented by a different state P|ψ>.

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[0206] When relying on Pauli-style frame tracking, a typical n-qubit state is, along with 2n classical bits describing the frame, 4 n It can be represented by any of the 100 possible physical quantum states. The use of stabilizer codes that protect the logical information effectively halves the number of bits required to describe the frame. A key characteristic of this technique is that most of the computations that can be described by Clifford operations can be performed independently of classical tracking information. Classical Pauli frame data only affects quantum operations performed at the logic level, such as magic state injection and distillation. This allows classical Pauli frame processing to be performed at the logic clock rate rather than the potentially much faster physical fusion clock rate. Below, we explain how Pauli frame tracking is naturally suited to fault-tolerant FBQC and why this technique imposes only minimal quantum and classical processing requirements.

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[0207] C. General Logic To achieve a universal gate set, state injection is added to the Clifford gates, which can be combined with the magic state distillation protocol and used to implement the T gate or other small-angle rotation gates. Magic state injection can be performed on single qubits by executing a modified fusion operation [Number] or by replacing the resource state with a special "magic" resource state. These approaches, along with the configuration of the injection sites, provide numerous ways to optimize the noisy encoded state preprocessing.

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[0208] D. Decoding and Other Clinical Processes In FBQC, classical error correction protocols are responsible for extracting reliable logical measurement information from unreliable and noisy physical measurement results, similar to other approaches to fault-tolerant quantum computing. In FBQC, it is useful to view the decoding result as logical Pauli frame information. Tracking this logical Pauli frame is necessary to interpret future measurement results.

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[0209] This logical Pauli frame generates time-sensitive information when logical level feedforward is required. That is, when logical measurement results are used to determine future logical gates, the relevant Pauli frame information needs to be made available. An example of this is the case of realizing the T gate by magic state injection, where S or S† is applied conditioned on the logical measurement result.

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[0210] One widely discussed challenge of decoding is that it must be performed live during quantum computation. However, a key feature is that this feedforward operation occurs on a logical timescale, and the decoding result is not required on the fusion (or physical qubit) timescale. If decoding is slower than the logical clock rate, buffering or auxiliary logical qubits can be used to allow the computation to "wait" for the decoding result. However, it is worth emphasizing that these are tools used at the logical level and do not require any modification of the physical operation. Fusion can always proceed without a decoding result. The important implication of this is that a slow decoder does not affect the threshold.

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[0211] Nevertheless, a high-speed decoder is desirable to reduce unnecessary overhead.

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[0212] E. FBQC Architecture For any given fusion network, there are many possible variations of the physical architecture.

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[0213] Figures 14A–14E show an example of a fusion router that provides routing to generate a 6-ring fusion network using a photonic resource state generator, optical routing, and linear optical fusion. This example demonstrates several features of the scheme for FBQC. 1. Resource state generators can be reused to create large fused networks. A fused network contains many resource states, but they do not all need to coexist simultaneously. Resource state generators (RSGs), which generate the state for each clock cycle, can be reused repeatedly. One natural way to create large fused networks in chronological order is to divide it into "time slices". A 3D fused network is divided into 2D layers, with one layer created at each time step and fused with the previous layer. This chronological ordering makes it possible to generate a 3D network using a 2D array of resource state generators. 2. The fused routing can be fixed so that rerouting between each clock cycle is not required. A good design principle in fused routing is to minimize the need for switching between clock cycles. This reduces switching losses and errors and minimizes the need to input classical control signals. In this exemplary layout, all resource states generated at a given location are directed to the same fused device. This means that the device connections are fixed and no switching is required to generate bulk. 3. Logic can be implemented by modifying the fusion measurements. To enable the logic to be implemented (at least), a subset of the fusion devices must be reconfigurable, as shown in Figure 14E. Boundaries or other topological features in the bulk are implemented by changing the fusion measurement basis or by switching to single-qubit measurements.

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[0214] The example in Figure 14 represents a simple physical architecture. There is considerable flexibility in how such architectures are constructed, particularly with photonic qubits. More extreme time-ordering techniques can be employed depending on how long resource states can wait in memory before incurring too much decoherence or loss. With photonic resource states, an entire block of a fused network can be created using a single RSG by applying the concept of interleaving, as described in International Publication No. 2020257772, to create the network with only one resource state at a time. Any time order is possible, as long as the time order of state creation is compatible by higher-level feedforward constraints at the logical level. This example includes only local connectivity, which is not a requirement for photonic qubits. Long-distance connectivity can enable the generation of codes embedded in non-periodic boundary conditions, other topologies, or non-Euclidean spaces. Higher-order fused networks can also be created by combining time-ordering structures with appropriate optical connectivity.

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[0215] 2.2.1. Circuit Symbols To facilitate understanding of the explanation, Figures 14A–14D introduce a set of schematic circuit symbols used in subsequent figures. These circuit symbols represent photonic / electronic circuits operating with physical qubits, where each input or output line represents a (physical) qubit. As a convention of the drawings, inputs are shown on the left and outputs on the right, with the separation that the schematic diagrams do not need to correspond to a specific physical layout.

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[0216] Figure 14A shows symbols representing a resource state generator (RSG) circuit 1400. As previously mentioned, an RSG circuit can be implemented using any photonic / electronic circuit that generates resource states. The output of the RSG circuit 1400 is a qubit, where each qubit is a line; the number of outputs depends on a particular resource state. In the embodiments described herein, it is assumed that the RSG circuit generates one resource state per clock cycle, and the length of the clock cycle can be defined based on the time required for one RSG circuit to generate one resource state. The required time may depend on a particular RSG circuit. For example, some existing RSG circuits can generate a resource state in about 1 ns, and the clock cycle can be 1 ns. In some embodiments, the clock cycle can be longer than the time required for the RSG circuit to generate one resource state, and the RSG does not need to operate at its maximum speed. For the purposes of this specification, it is assumed that the RSG circuit 1400 outputs all qubits of a resource state in the same clock cycle. However, as will be apparent to those skilled in the art accessing this disclosure, the timing can be changed.

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[0217] Figure 14B shows a symbol representing a Type II fusion circuit 1405. The Type II fusion circuit can be implemented as described below with reference to Figures 23 to 36, for example, and may be reconfigurable as previously mentioned with reference to Figure 3D. The input is two qubits consumed by the Type II fusion operation as described below (indicated by solid lines with inward arrows). The Type II fusion circuit 1405 can provide a classical output signal 1406 indicating the measurement result of the fusion and the success or failure of the fusion operation and / or a particular type of success or failure (e.g., a pattern of detected photons), where the pattern of detected photons indicates the number of photons detected by each of the detectors of the fusion circuit.

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[0218] Figure 14C shows symbols representing the switching circuit 1410. The inputs and outputs to the switching circuit 1410 can contain any number of qubits, and the number of inputs does not have to be equal to the number of outputs. The switching circuit 1410 can incorporate any combination of one or more active optical switches, mode couplers, phase shifters, etc. The switching circuit can be configured to perform active operations that reconfigure the input modes (e.g., to bring about a ground state change of qubits by coupling the modes of qubits) and / or to apply a phase to one or more of the input modes (which may affect subsequent coupling between modes). In some embodiments, the operation of the switching circuit 1410 can be dynamically controlled in response to a classical control signal 111, and its state can be determined based on the results of previous operations, specific calculations performed, configuration settings, timing counters (e.g., for periodic switching), or any other parameters or information.

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[0219] Figure 14D shows symbols representing a delay circuit 1415. A delay circuit can delay a qubit for a certain period of time and function as a memory for the quantum information stored in the qubit. The length of time (in clock cycles) is indicated by a number; in this example, +1 means a delay of 1 clock cycle. In the case of photonic qubits, the delay circuit can be implemented, for example, by providing one or more optical fibers or other waveguide materials of appropriate length so that photons of the delayed qubit travel a longer path than photons of the non-delayed qubit.

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[0220] Figure 14E shows a schematic diagram of a system for FBQC using what is referred to herein as a networked RSG circuit, according to several embodiments, such a circuit can generate the topological features described above to implement fault-tolerant quantum logic gates. The circuit notation is as described above in relation to Figures 14A to 14F, except that classical inputs and outputs are not shown for clarity in the diagram. Figure 14A shows a system using four representative network cells 1400, 1400', 1400'', and 1400'''. Figure 14A also shows coupling between adjacent instances of network cell 1400 in the network, such that the coupling forms an example of a fused network router that implements quantum error correction codes as described above. Each network cell 1400 may include an RSG circuit 1402 that generates a resource state having six peripheral qubits (e.g., the six-ring resource state 1410 shown here). RSG1402 provides two qubits to an adjacent network cell, as indicated by the output path of qubit 5, referred to herein as the "x-fusion direction," and the output path of qubit 6, referred to herein as the "y+ fusion direction." Network cell 1400 also receives qubits from two adjacent network cells. Specifically, the output path of qubit 2, referred herein as the "x+ fusion direction," is coupled in the fusion circuit to the output path of output qubit 5' of the adjacent network cell 1400''. Similarly, the output path of qubit 3 is coupled to the output qubit path of qubit 6'' of network cell 1400''.

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[0221] Network cells 1400, 1400', 1400''', 1400'''' may also include reconfigurable fusion circuits, e.g., reconfigurable fusion circuit 1420', and thus quantum logic can be implemented as shown in Figures 13A to 13D. Any or all of the other fusion circuits may be reconfigurable depending on the architecture being implemented, as will be described in more detail with reference to Figure 3D. Furthermore, offsets can be implemented as shown to enable fusion between qubits in resource states generated in different clock cycles. For example, in each RSG shown, qubit 1 from a resource state generated in the first clock cycle is fused with qubit 4 from a (different) resource state generated in a subsequent clock cycle. Delays other than the +1 clock circuit can be implemented and / or the delays can be applied to other qubits to implement fusion between any layers, and also, for example, the interleaving strategy described in International Patent Application Publication No. 2020257772A1, the entire contents of which are incorporated herein by reference in their entirety for any purpose.

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[0222] Figures 15A and 15B show examples of networked RSG circuits that can be used in material-based qubit architectures such as trapped ions and superconducting qubits, according to several embodiments. These embodiments also use six-ring resource states as shown, where qubits are included as numbered circles. Solid lines indicate qubit couplings that can perform entanglement operations between qubits. For example, such gates can perform a two-qubit bell projection measurement by performing a CZ gate between two qubits, followed by performing two single-qubit measurements in the computational basis. For example, in 15A, resource states are first generated by applying a CZ gate between the following pairs of qubits, namely (1,6), (1,2), (2,3), (3,4), (4,5), and (5,6), in order to perform an FBQC protocol similar to that described elsewhere in this specification. Then, two-qubit projection bell measurements are applied between resource states, for example by performing a two-qubit measurement (fusion) on the qubit pairs (2,5), (3,6), and (4,1'). Next, the state of qubit 1 is teleported to qubit 1'. In the next time step, a new resource state is generated as before, and the process is repeated.

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[0223] In the embodiment shown in Figure 15B, the FBQC protocol proceeds as follows: In step 1, the 6-ring resource states are preprocessed as described above by applying CZ gates between appropriate pairs of qubits: (1,6), (1,2), (2,3), (3,4), (4,5), and (5,6). In step 2a, fusion is applied between resource states, for example, between qubits (2,5) and (3,6) from adjacent (different) resource states, and between (4,1') within each resource state. Next, in step 2b, the state of qubit 1 is teleported to qubit 1' within each resource state.

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[0224] F.FBQC error The thresholds presented in this specification are based on a simple error model. In physical embodiments, there are many things that can affect the performance of the system. The error channel can have much more structure than random i.i.d. Pauli errors including error bias and correlations, and the chronological ordering of operations can spread errors. When logical gates are implemented by creating topological features such as boundaries or twists, these require different physical operations and result in different error models at their locations.

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[0225] However, nonetheless, there are several reasons why the results presented here can still be significant across many types of physical hardware. · Resource states and fusion errors are essentially local. The construction of FBQC limits how far errors can potentially spread. Correlations are expected to exist only within resource states and not between resource states. This expectation is particularly strong in linear optics where photons at different locations do not get "accidentally" entangled with each other. Further, each qubit in the protocol has a short finite lifetime, limiting the potential for error diffusion in its vicinity. Further, since the resource states and fusions in the inventors' model are all the same, it is a reasonable assumption that they have the same or similar error rates in physical embodiments. · Correlations within fusions can only improve performance. The places where correlated errors can occur are between the two measurement results of the fusion operation. The inventors' model treats these errors as uncorrelated. Since the inventors here decode the primal and dual syndrome graphs separately, if the fusion errors are correlated, it makes no difference to the inventors' threshold. If there is a way to account for that information, it can only improve performance. · The bulk determines the threshold. The topological features used to implement the logic are two - dimensional or one - dimensional objects. As a result, the error threshold is determined by the bulk.

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[0226] Despite the complexity of evaluating the entire system, many hardware-level error models can be adequately approximated by our model via error channel remapping. For example, under a standard gate error model, if a resource state was to be constructed from a series of noisy 2-qubit gates bearing an error of probability p_physical, then Pauli-X and Pauli-Z accumulated by each qubit during state preprocessing can be considered and re-expressed as the cumulative error rate. When a cluster state is constructed from 2-qubit gates, the error cannot propagate further than their nearest neighbor, and therefore the correlation from the propagating error is only between the primal and the dual.

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[0227] X. Discussion At the level of fault-tolerant logic gates, fusion-based quantum computing enables the same operation as circuit-based quantum computing (CBQC) or measurement-based quantum computing (MBQC). However, when we look at the physical processes used to implement the logic gates, significant differences emerge in the resource dependencies required for the computation protocol, the necessary connectivity of physical qubits, classical information processing, and the emergence, propagation, and impact of errors. One difference between FBQC and MBQC lies in the nature of the respective entangled states required to implement fault tolerance. MBQC requires large entangled cluster states that scale up with the computation being performed, while FBQC requires constant-size resource states, the number of which increases with larger quantum computations. This distinction is also evident in the type of measurement used. MBQC uses single-qubit measurements to perform computations, and previous research proposing the LOQC architecture to achieve fault-tolerant MBQC was done by first creating large cluster state resources from finite-size computations and then using computational (single-qubit) measurements. Such separation does not exist in the FBQC protocol, where multi-qubit projection measurements, such as fusion gates, integrate the entanglement measurements necessary to generate long-range entanglement with the measurements that implement fault tolerance and computation. Although not mandatory, in some variants of FBQC, the protocol can also include a small number of single-qubit measurements to create topological features, such as those shown in Figures 13A–13D.

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[0228] Regarding fault-tolerant computations using linear optics, more distinctions arise at the structural level. LOQC has a long history, and the earliest proposals relied on very large gate-long transitions or repeat-failure strategies to handle stochastic gates, requiring quantum memory. More recent architectural proposals eliminated the need for memory, and the schemes for fault tolerance were based on constructing large entangled cluster states and then performing single-qubit measurements on those states to perform fault tolerance and computation via MBQC. These schemes have a low constant depth, meaning that each photon sees a small, fixed number of components during its lifetime, regardless of the size of the computation. The most high-performing schemes were based on percolation methods to handle stochastic fusion. Other schemes could tolerate stochastic fusion at the expense of reducing the threshold for loss and Pauli error by using branched resource states to add redundancy.

[0193]

[0229] The FBQC scheme disclosed herein uses a fixed size of resources in an architecture of a fixed depth, but provides a significant threshold improvement compared to the best results in the literature. Beyond the error tolerance, FBQC offers a significant advantage in architectural feasibility compared to those earlier schemes, which required classical processing and feedforward to occur during the lifetime of a photon. This classical processing is often complex, and the need to perform this kind of global computation while photons are waiting in delay lines imposes extraordinary requirements on the loss of photon delay. FBQC eliminates this requirement, making the fault tolerance threshold independent of the timescale of classical feedforward. According to some embodiments, feedforward is still required, as in the case of any quantum computing architecture, but in FBQC, this requirement is only at the logical level, and the timescale is completely isolated from the physical calculations.

[0194]

[0230] FBQC is a modular architecture composed of small, distinct functional blocks: resource state generators, fusion networks, and fusion operations. While the blocks need to be compatible with each other, the physical implementation of each block can be independent, and in practice, there are multiple options. As an example of its application to a realistic physical system, we present a fully linear optical implementation of FBQC. However, it is more generally applicable to other physical platforms, and in particular, critically supports applications in hybrid quantum systems. For example, photonic fusion operations and fiber-based fusion networks can be integrated with material-based resource state generators that generate photonic entangled states. Modularity is also a crucial aspect that ensures the architecture of quantum computing is reliable and manufacturable.

[0195]

[0231] XI. Decoder Buffering The challenges of decoding and handling classical processing and feedforward at the logic level are shared across all models of quantum computing. Decoders are highly unlikely to operate as fast as the physical clock speed of the quantum processor. Classical feedforward is required at the logic level (see Figure 16), and therefore, decoder systems should be designed to handle long latency and improve decoder throughput. Here, decoder buffering and decoder parallelization are described as techniques to address these problems.

[0196]

[0232] Figure 16 shows an example of a quantum circuit that requires logic feedforward. More specifically, Figure 16 shows a circuit for performing a logic π / 8 rotation. Pauli product measurement.

number

number

[0197]

[0233] A. Decoder System First, we will introduce the basic functionality of the decoder system and how it interacts with the quantum processing of logic qubits and other classical processing. A schematic diagram of the classical information entering and leaving the quantum system is shown in Figure 17. This diagram shows the system's evolution over time when two logic gates are implemented. Here, we consider an example of topological quantum computing where a 2D layer of measurement information is acquired at each time step. Physically, this can be achieved in either an FBQC setting where fused measurements are performed on resource states within a 3D fused network, or in a 2D surface code where parity check measurements are performed at each time step. This system includes three subsystems. The quantum processor 1701 contains quantum information. The quantum processor can receive and execute measurement instructions, which enable the implementation of fault tolerance and logic gates. The logic gate control system 1703 includes a program for the quantum algorithm to be executed. This includes instructions for logic gates and feedforward instructions on which gates to implement based on previous measurement results. It receives the output from the decoder and sends the logic gate instructions to the quantum processor. The decoder system 1705 receives measurement information from the quantum processor and decodes it by performing classical calculations. The decoder system transmits its output to the logic gate control system.

[0198]

[0234] B. Information Flow The flow of information can be understood using the following steps, which are indicated by numbered circles in the diagram.

[0199]

[0235] In step 0, the logic gate control includes a quantum program. This originates from user input and can be compiled offline before execution by the quantum processor. The quantum program may include feedforward steps where future instructions depend on measurements performed by the quantum system. After several steps of the program have already been executed, the logic gate control has a current program state. 1. First, an instruction is issued by the logic gate control system and sent to the quantum processor. 2. Instructions are executed on a quantum system. To implement a logic gate, this may involve executing multiple layers of instructions over L time steps. Depending on the nature of the quantum hardware, instructions can be gate sequences, fused measurement patterns, single-qubit measurements, or other physical quantum instructions. 3. After the logic gates are executed, the L layer of measurement information is accumulated, which is then bundled together and passed to the decoder. 4. The decoder receives the measurement information, which is called the decoding problem, and calculates the corrected measurement result. 5. Once decoding is complete, the result is returned to algorithm control and used to calculate which logic gate instruction should be issued next.

[0200]

[0236] G. Timescale There are several timescales related to determining how the system should be configured to ensure that logic gate instructions are available when needed. • Layer clock time - This is the time between each layer of computation. c This timescale can vary significantly between different physical systems. In a photonic system without interleaving, this can be as fast as 1 ns. With interleaving, it can be reduced to less than 1 μs. · Logical block time - The time to implement logic gates across L layers, which takes time t_log = L * t_c. The value of L required to reach the target logical error rate typically ranges from 30 to 50. As an example, take the value L = 40. Depending on the level of interleaving applied, t_log can be as low as about 40 ns or greater than 40 μs. · Decoder latency - The time it takes to execute the decoder, t D Here, this latency is the time from the final measurement in the quantum system until the first logical instruction can be executed, i.e., the time for signal transmission to / from the quantum system and the time for calculation of algorithm instructions. The latency cannot be reduced by parallelization. This timescale is indicated by the thick arrow in Figure 17. The decoder runtime depends on the system size, error rate, and decoding algorithm. Since the runtime also depends on the specific measurement configuration, in practice, the time varies for each execution according to several distributions.

[0201]

[0237] We can consider three different regimes of these timescales that require different configurations and methods of the decoding system. 1. Immediate decoding: t D < t c - In the simplest scenario, the decoder evaluation completes within 1 layer clock cycle. In this case, the logical instruction of gate 2 is available within the time for the gate to be executed without delay immediately after gate 1. 2. Fast decoding: t_c < t_D < t_log - In the second scenario, the decoder is slower than 1 clock cycle but completes faster than the time it takes to execute a complete logic gate. In this case, there must be a delay between the completion of logic gate 1 and logic gate 2. However, in this logical qubit, since the execution of the decoding problem of logic gate 1 is completed by the time it takes to start decoding logic gate 2, only one decoder processor is required. In other words, the decoder throughput is large enough to keep up with the rate at which information is generated. 3. Slow Decoding: t_c > t_log - Finally, consider the case where the decoder execution time is longer than the logic gate time. In this case, there must be a delay between gates, but further problems arise with the decoder throughput. With one decoder processor, the first decoding problem is still running when the second decoder processor arrives. This leads to a "backlog problem" where latency inherently increases with each subsequent gate as the queue of pending decoding problems grows. Fortunately, this throughput problem can be solved by using multiple decoder processors to increase throughput to match that of a quantum system.

[0202]

[0238] The "instantaneous decoding" scenario is highly unlikely to be achievable in quantum computers, and the need to address decoder delays and parallelize decoders will almost certainly be encountered. The following section introduces the concept of decoder buffering, which can be used as a technique to handle these decoding timing scenarios.

[0203]

[0239] H. Decoder system designed to handle slow decoding To address the problems arising in the "fast decoding" and "slow decoding" regimes defined in the previous section, a combination of modifying the logic circuits to allow decoding delays and adding additional processors to improve throughput can be used.

[0204]

[0240] A. Decoder buffering If the decoder latency is longer than the layer clock time, decoder buffering can be used to allow the target logic qubit to wait until the next logic gate instruction is available. This buffer region simply implements the identity region where the measurement result is needed for feedforward after each logic gate. This identity operation has no logical effect on the qubit. Importantly, the identity "buffer region" does not need to be decoded in order to determine the next logic gate measurement instruction. The buffer region needs to be decoded at a later point in time, but the result of that decoding updates the Pauli frame used to interpret future measurement results, but this does not change the determination of what the logic measurement instruction is.

[0205]

[0241] The buffer time can be selected as a fixed duration before each feedforward operation, long enough to cover the entire decoding execution time. Alternatively, the duration of the buffer area can be adaptively selected so that the logical qubits wait in memory until the next gate instruction becomes available.

[0206]

[0242] B. Parallelization of the decoder If the logical latency is slower than the logical clock speed, in addition to the buffer, an additional decryption processor may be included to increase throughput so that decryption can be performed at a speed that can "catch up" with the information being generated.

[0207]

[0243] C. Exemplary Decoding System In Figure 18, the inventors show an exemplary configuration of a decoding system that includes both buffering and decoder parallelization. Consider an example of logic feedforward required for a magic state injection circuit. In this case, the first logic gate is a state injection that includes coupling a target logic qubit with a distilled magic state and performing a logic measurement. The second logic gate is a correction circuit that is executed in response to the logic result of the first gate. The example in Figure 18 shows a case where the decoder runtime is approximately twice the logic clock time and the total decoding delay is approximately three times the logic clock time. A buffer with a duration longer than the latency is added so that correction gate instructions are preprocessed by the end of the buffer region. To improve decoder throughput to match the quantum system, two decoder processors are used per logic qubit, and the decoding problem is then passed to them alternately.

[0208]

[0244] XII. Physical Level Components and Operation Figure 19 shows photonic hardware components in a linear quantum computer according to several embodiments. In a photonic chip for linear quantum computing, a single photon is emitted by a photon source, passes along a waveguide through a series of linear optical elements including a delay, a directional coupler, and an active phase shifter, and is then detected by a photon counting detector.

[0209]

[0245] In some embodiments, qubit state initialization includes a single-photon source. If the photon source is successful, it produces only one photon. The photons produced by each photon source are substantially identical, including frequency, pulse shape, and timing. In some embodiments, the photon source can produce photons at a very high repetition rate, e.g., about 1 GHz. A suitable photon generation technique includes spontaneous four-wave mixing to stochastically generate photon pairs at mid-infrared frequencies close to the optical communication band. One of the two photons is detected and generates an electrical signal indicating the source's success.

[0210]

[0246] Since spontaneous four-wave mixture sources do not function at a unit probability, it is desirable to multiplex them. By multiplexing multiple sources that operate at a low probability, it is possible to generate a single source that operates at a high probability.

[0211]

[0247] In multiplexed sources, it may be desirable to delay the generated single photon for, for example, 1 or 2 nanoseconds. This delay provides time for the Herald detector to emit and for the logic necessary to activate the optical switch to execute. The switch routes the photon from the successful source to the desired output waveguide.

[0212]

[0248] According to some embodiments, delay is generated using ultra-low-loss waveguides. Some architectures may utilize optical fibers, which can enable longer delays. It should be noted that the delay used in FBQC architectures is short and fixed (i.e., it does not increase as the size of the computation increases).

[0213]

[0249] According to several embodiments, the optical switch can be implemented using a generalized Mach-Zehnder interferometer (GMZI). These interferometers consist of an array of active phase shifters sandwiched between two perfectly mixed interference networks (e.g., Hadamard networks). An active phase shifter is a device that can achieve optical phase shift when an applied voltage is applied. The perfectly mixed interference networks can be implemented using passive linear optical elements. The main feature of these interference networks is that they convert any single-photon input into a wave function that is uniformly diffused across all modes. At this point, each mode enters an active phase shifter that performs one of two phases (0 or π) to an optical mode. After this, the mode enters another perfectly mixed interference network. This embodiment allows routing any input mode of the optical switch to any output mode. There is only a single active phase shifter in the photon path, minimizing losses in the switching network. If it is only necessary to switch N input modes to a single output mode, the second interference network can be greatly simplified.

[0214]

[0250] According to several embodiments, a photon-number-resolved detector can perform qubit measurements, detect herald photons emitted by a source, and perform the measurements necessary to generate resource states. While many such detectors can be used, the chosen technique should have very high quantum efficiency so that photons are detected with very high probability when they collide with the detector. The detector should also have a low dark count so that the probability of the detector emitting light without incident photons is very low in each time bin. The detector must also be number-resolved so that two counts are reported with high probability when two photons collide with the detector. Finally, to operate with a 1 GHz single-photon source, the detector should have very low timing jitter and a fast reset time.

[0215]

[0251] According to several embodiments, superconducting nanowire single-photon detectors (SNSPDs) can be used as a preferred single-photon detection technique for near-infrared photons. The combination of speed, timing accuracy, and detection efficiency is superior to many alternatives, although any detector technique can be used without departing from the scope of this disclosure. In general, SNSPDs require cryogenic operation at several Kelvin (considerably higher than the millikelvin temperature of many material-based qubits). Furthermore, the design of a number-resolved detector can include multiple SNSPDs. One conceptually simple way to achieve this is by fanning out the incident waveguides into a bank of SNSPDs, although other designs are possible without departing from the scope of this disclosure. The number of SNSPDs should be such that the probability of two incident photons colliding with a single SNSPD is sufficiently low.

[0216]

[0252] According to some embodiments, these hardware components (source, detector, delay, and switch) reside within a multiplexed single-photon source, as shown in Figure 20.

[0217]

[0253] Each photon source receives a pump laser input pulse and generates photon pairs, signal photons, and herald photons. The herald photons can be injected into a bank of photon-number-resolved detectors. The signal photons pass through a delay before being transmitted to the optical switching network. The schematic diagram shows a GMZI with six input modes and a single output. The Hadamard network is a network of directional couplers that implement the Hadamard transform for the input modes. In Figure 20, for simplification, only the optical elements are shown; the electronic components and interconnects for logic and feedforward are not shown.

[0218]

[0254] In each time bin, the electrical signal from each detector passes through some classical logic that determines which photon source produced the photons. The signal from the logic unit activates the phase shifter in the GMZI. These electrical components are not shown in the schematic diagram. The optical delay must be long enough to allow the detection, logic, and activation of the phase shift to take place.

[0219]

[0255] Essentially, the same approach to multiplexing can be used at several stages of the architecture. In one example, a typical resource state generator takes multiplexed single photons as input and then generates Bell or GHz states using standard methods from the literature. These relatively small entangled states are referred to herein as seed states. The resource state generator requires a supply of multiplexed seed states to be used to construct larger entangled resource states through fusion. According to some embodiments, the fusion step itself can be multiplexed.

[0220]

[0256] The overhead associated with multiplexing depends on the success probability required for the single-photon source, seed state, or resource state. Preferably, in an FBQC architecture, the success probability of the single-photon source can be chosen independently of the size of the computation. The same applies to both seed state generation and resource state generation. This is partly because if single-photon generation, seed state generation, or resource state generation fails, a qubit will eventually be lost at a known location, i.e., these errors will be announced. Error correction codes can correct these missing qubits as long as the system remains below the error correction threshold.

[0221]

[0257] XIII. Further Embodiments Figure 21 shows a possible example of a fusion region 6001 configured to work with a fusion controller 319 to provide measurement results to a decoder for fault-tolerant quantum computing, according to several embodiments. In this example, the fusion region 6001 can be an element of a fusion array 321 (shown in Figure 3), and although only one example is shown for illustrative purposes, the fusion array 321 can contain any number of examples of the fusion region 6001.

[0222]

[0258] As previously stated, the qubit fusion system 305 can accept two or more qubits to be fused (qubit 1 and qubit 2, shown here in dual-rail coding). Qubit 1 is one qubit entangled with one or more other qubits (not shown) as part of a first resource state, and qubit 2 is another qubit entangled with one or more other qubits (not shown) as part of a second resource state. Preferably, in contrast to MBQC, to facilitate fault-tolerant quantum computation, none of the qubits from the first resource state need to be entangled with any of the qubits from the second (or any other) resource state. Also preferably, at the input to the fusion site 6001, the set of resource states do not entangle with each other to form a cluster state in the form of quantum error-correcting codes, and therefore there is no need to store and / or maintain a large cluster state with long-range entanglement across the entire cluster state. Preferably, the fusion operation performed at the fusion site can be a completely destructive, batch measurement of qubit 1 and qubit 2, and all that remains after the measurement is classical information representing the measurement results at detectors, e.g., detectors 6003, 6005, 6007, and 6009. At this point, the classical information is all that the decoder 333 needs to perform quantum error correction, and no further quantum information is propagated through the system. This may be in contrast to MBQC systems that use a fusion site to fuse resource states into cluster states that themselves function as topological codes, and then generate the necessary classical information through single-particle measurements for each qubit in the large cluster state. In such MBQC systems, not only is it necessary to store and maintain the large cluster state in the system before single-particle measurements are performed, but an additional single-particle measurement step (in addition to the fusion used to generate the cluster state) must be applied to all qubits in the cluster state to generate the classical information necessary for the decoder to compute the syndrome graph data required to perform quantum error correction.

[0223]

[0259] Figure 21 shows an exemplary example of one method of implementing a fusion site as part of a photonic quantum computer architecture. In this example, qubit 1 and qubit 2 may be dual-rail coded photonic qubits. A brief introduction to dual-rail coding of photonic qubits is provided in Section XIV below, with reference to Figures 26A to 29. Thus, qubit 1 and qubit 2 can be input to waveguide pairs 6021, 6023 and waveguide pairs 6025, 6027, respectively. Interferometers 6024 and 6028 can be aligned with each qubit, and within one arm of each interferometer 6024, 6028, programmable phase shifters 6030, 6032 can be optionally applied to implement, for example, a specific mode coupling as shown in Figure 21, which is referred to herein as XX, XY, YY, or ZZ fusion, thereby influencing the basis to which the fusion operation is applied. The programmable phase shifters 6030 and 6032 can be coupled to the fusion controller 319 via control lines 6029 and 6031 so that signals from the fusion controller 319 can be used to set the basis to which the fusion operation is applied to the qubits. In some embodiments, the basis can be hardcoded within the fusion controller 319, or in some embodiments, the basis can be selected based on instructions provided by an external input, e.g., the fusion pattern generator 313. Additional mode couplers, e.g., mode couplers 0633 and 6032, followed by single-photon detectors 6003, 6005, 6007, and 6009 can be applied after the interferometer to provide a readout mechanism for performing batch measurements.

[0224]

[0260] In some embodiments, the fusion may be a probabilistic operation, i.e., a probabilistic Bell measurement may be performed, which may or may not be successful, as shown in Figure 35 below. In some embodiments, the success rate of such an operation may be increased by using an additional quantum system in addition to the one on which the operation is acting. Embodiments that use an extra quantum system are typically referred to as a “boosted” fusion. As those skilled in the art will see, any type of fusion operation can be applied (which may or may not be boosted) without departing from the scope of this disclosure. Additional examples of Type II fusion circuits are shown and described in Section XIV below for both polarization coding and dual-rail path coding. In some embodiments, the fusion controller 319 may also provide control signals to detectors 6003, 6005, 6007, and 6009. The control signals may be used, for example, to gate the detectors or otherwise to control the operation of the detectors. Each of the detectors 6003, 6005, 6007, and 6009 provides a photon detection signal (representing the number of photons detected by the detector, e.g., 0 photons detected, 1 photon detected, 2 photons detected, etc.), which can be preprocessed at the fusion site 6001 to determine the measurement result (e.g., success or failure of fusion) or passed directly to the decoder 333 for further processing.

[0225]

[0261] Example of FBQC using GHZ resource state

[0226]

[0262] Figures 22A and 22B illustrate FBQC schemes for fault-tolerant quantum computing according to one or more embodiments. In these examples, a topological code known as a Raussendorf lattice (also known as a leaf code) is used, but any other error-correcting code can be used without departing from the scope of this disclosure. For example, FBQC can be implemented in various volume codes (e.g., diamond codes, etc.), various color codes, or other topological codes without departing from the scope of this disclosure.

[0227]

[0263] Figure 22A shows one unit cell 2202 of the Raussendorf lattice. For measurement-based quantum computing, P is located at the center of the unit cell as specified herein. cell To determine the values ​​of the syndrome graph called, the qubits on the six faces of the unit cell are measured in the x basis, and as a result, six M x For each measurement, a set of zero or one eigenvalue is determined. These eigenvalues ​​are then combined as follows:

number

[0228]

[0264] Here, S1, S2, ..., S6 correspond to six parts on the surface of the unit cell, M x (S i ) corresponds to the measurement result (0 or 1) obtained by measuring the corresponding face qubit with the x basis. (In Figure 22, S1, S2, and S3 are labeled. S4, S5, and S6 are located on the hidden faces of unit cell 2202.)

[0229]

[0265] In FBQC, the goal is to generate a set of classical data corresponding to an error syndrome of some quantum error correction code through a series of batch measurements (e.g., a positive operator value measure, also known as POVM) on two or more qubits. For example, using the Raussendorf unit cell in Figure 22A as an exemplary example, Figure 22B shows a set of measurements that can be used to generate syndrome graph values ​​in the FBQC approach. In this example, the GHZ state is used as the resource state, but as those skilled in the art who benefit from this disclosure will see, any suitable resource state can be used without departing from the scope of this disclosure. To convert the MBQC scheme shown in Figure 22A to the FBQC scheme shown in Figure 22B, each face qubit in Figure 22A is replaced with individual qubits from clearly separated (i.e., unentangled) resource states. For example, the four resource states R1, R2, and R3 (enclosed by dotted ellipses) each contribute at least one qubit to what becomes the face qubit S2 of the Raussendorf cell, and are labeled in Figure 22B. For example, in Figure 22A, the face qubit S2 is replaced with 4 qubits from three different resource states, where resource state R1 contributes to 2 qubits, resource state R2 contributes to the third qubit, and resource state R3 contributes to the fourth qubit. During operation, the system performs two fusions on each face (for example, circles 2221 and 2222 in Figure 22B represent fusions between resource states R2 and R1 and between the contributing qubits of R3 and R1, respectively). In the example where the fusion is a type II fusion, all four face qubits are measured, thereby generating four measurement results. The syndrome graph value for the cell is obtained by equation (2) above, but here, M x (S i )=[F 1,XX (S i )+F 2,XX (S i )]mod2 (3)

[0230]

[0266] Here, for the i-th face, F 1,XX (S i ) is a measurement result obtained by performing a batch measurement on the qubit associated with fusion 1 (e.g., as shown by circle 721), where fusion 1 is a type II fusion performed on an XX basis, F 2,XX (S i ) is a measurement result obtained by performing a batch measurement on the qubit associated with fusion 2 (e.g., as shown by circle 722), and fusion 2 is also a type II fusion performed on an XX basis. Similar to the measurements associated with the observable X mentioned above in relation to equation (2), the observable fusion measurements of XX (and ZZ) take values ​​of 0 or 1, corresponding to the positive or negative eigenvectors of the measured operators (XX and ZZ, in this example), respectively. Considering equation (3), plane M x (S i To obtain each of the above measurements, fused measurement F 1,XX (S i ) and F 2,XX (S i A correct fusion result for both ) is desirable. However, if either fusion fails due to some error and the operator value cannot be recovered, in some embodiments the face measurement is considered a failure, resulting in at least one erased edge of the syndrome graph data. As will be seen by those skilled in the art who are interested in this disclosure, the error can be dealt with by a decoder in a manner similar to that described above with reference to Figures 1A-1C. As will be seen by those skilled in the art, although our description of equation (3) focuses on the XX observable, the fusion can also generate measurements of the ZZ observable, and their results can be combined as in equation (3) to generate independent syndrome graph date sets. In some embodiments these two sets of syndrome data are referred to as primal and dual syndrome graphs.

[0231]

[0267] Figure 22C shows an example of a cluster state constructed on several unit cells of a Raussendorf lattice. The MBQC method requires generating this entire cluster state, forming an entangled state of many qubits with entanglement of states extending across the lattice from one surface boundary to another. In the MBQC method, it is this large entangled cluster state that functions as a quantum error correction code and is therefore capable of encoding logical qubits. The computation proceeds by performing single-qubit measurements on each qubit of the entangled state to generate measurement results used to generate a syndrome graph supplied to the decoder, as described above with reference to Figures 1A-1C. Thus, increasing the error tolerance of the computation requires increasing the size of the lattice, and therefore increasing the size of the entangled state. In one or more embodiments of the FBQC method disclosed herein, such a large entangled cluster state is not required; rather, a smaller resource state is generated, and the size of the resource state is independent of the required error tolerance. As described in detail above with reference to Figure 22, the FBQC method can be constructed from any fault-tolerant grid by replacing each node of the grid with a set of fusions between two or more adjacent resource states. This configuration, in which each node of the grid is replaced with a resource state / fusion unit, is just one example of how to obtain an FBQC method, and as those skilled in the art who benefit from this disclosure will see, many different methods can be employed to construct an FBQC method from a fault-tolerant grid without departing from the scope of this disclosure.

[0232]

[0268] Furthermore, as will be explained in more detail below, the process can be carried out by generating layers of resource states in a given clock cycle and performing fusion within each layer, as illustrated in Figures 23-24 below. For example, in Figure 22C, the horizontal direction represents time, meaning that all or a subset of qubits in any given layer in the xy plane can be generated / initialized in the same clock cycle, for example, qubits in layer 1 can be generated in clock cycle 1, qubits in layer 2 can be generated in clock cycle 2, qubits in layer 3 can be generated in clock cycle 3, and so on. As will be explained in more detail below, a specific subset of qubits in each layer can be stored / delayed so that they are available to be fused with qubits from resource states in subsequent layers, if necessary, to enable fault tolerance.

[0233]

[0269] In some embodiments, a lattice preprocessing protocol (LPP) can be designed to generate a suitable syndrome graph from the fusion of multiple smaller entangled resource states in order to produce a desired error syndrome. Figures 23-24 show examples of lattice preprocessing protocols according to some embodiments. For illustrative purposes, the resource states are states such as resource state 2300 shown in Figure 23A, but other resource states can be used without departing from the scope of this disclosure. Resource state 2300 is equivalent to a GHZ state up to the application of a Hadamard gate to a single qubit. For example, the state used in the examples disclosed herein is equivalent to a GHZ state up to the application of a Hadamard gate to the two terminal qubits 2300a-3 and 2300a-4 in Figure 23A. More specifically, a 4-GHZ state can be identified as a stabilizer state having the following stabilizer: <XXXX,ZZII,ZIZI,ZIIZ>. The resource state 2300 shown in Figure 23A is closely related to this GHZ state, but the stabilizer for state 2300 is <XXZZ,ZZII,ZIXI,ZIIX> (the order of the operators corresponds to qubits 2300a-1, 2300a-2, 2300a-3, and 2300a-4, respectively). As those skilled in the art will see, the 4-GHZ state and resource state 2300 are equivalent under the application of Hadamard gates on qubits 2300-a3 and 2300-a4.

[0234]

[0270] The time direction in Figures 23-24 is perpendicular to the page, so that resource states having a shape such as resource state 2310 represent a set of qubits 1, 2, and 3 entangled with each other within the same clock cycle, and qubit 4 entangled with qubits 2 and 3 in the time dimension. Such resource states can be generated, for example, by generating a complete 4-qubit resource state in a single clock cycle and then storing qubit 4 in memory over a fixed period (e.g., one clock cycle). As used herein, the term “memory” includes any type of memory, e.g., quantum memory, qubit delay lines, shift registers for qubits, qubits themselves, etc. In the case of photonic resource states, such qubit memories are equivalent to qubit delays and can therefore be implemented using optical fibers. In the example shown in Figure 23C, the delay up to qubit 4 is schematically represented by a loop of additional optical path length (e.g., provided by an optical fiber) that is inline with the existing optical path of the qubit but not present in the optical paths of qubits 1-3. In this example, the fiber length is such that it implements a single clock cycle delay of duration T, but other delays, such as 2T, 3T, etc., are also possible. With respect to the physical delay time, such delays may be in the range of 500ps to 500ns, but any delay is possible without departing from the scope of this disclosure.

[0235]

[0271] Returning to the FBQC process disclosed herein, Figures 23-24 illustrate an example of how the FBQC lattice preprocessing and measurement protocol can proceed layer by layer. Figure 23A shows a portion of the lower layers of the Raussendorf lattice, shown as layer 2310 (corresponding to a portion of layer 1 shown in Figure 22C). In this example, a number of initial resource states 2300 are generated to process a layer such as that shown in Figure 23A (for example, in the qubit entanglement system 303 in Figure 3). In this example, the resource state 2300 is an entanglement state containing four physical qubits (also referred to herein as quantum subsystems), namely qubits 2300a-1, 2300a-2, 2300a-3, and 2300a-4. In some embodiments, the resource state 2300 can take the form of a 4-GHz state in which two terminal qubits 2300a-4 and 2300a-3 have undergone an Hadamard operation (for example, in the case of a dual-rail coded qubit, by applying a 50:50 beam splitter between the two rails forming the qubit). In some embodiments, not all qubits in a layer undergo fusion in this clock cycle; rather, some of the qubits generated from a particular resource state during this clock cycle can be delayed, for example, by delaying the measurement of qubit 2320, the redundant coded qubit 2305, or any other qubit so that the qubit becomes available in the next clock cycle. Such delayed qubits are then available to fuse with one or more qubits from the resource state that are only available for fusion in the next clock cycle.

[0236]

[0272] In an example using a photonic implementation, the qubits from the resource state are then appropriately routed (via an integrated waveguide, optical fiber, or any other suitable photonic routing technique) to a qubit fusion system (e.g., qubit fusion system 305 in Figure 3) to enable a set of fused measurements that implement quantum error correction, i.e., to collect measurement results corresponding to a selected error syndrome. This example explicitly uses a topological code based on a Raussendorf lattice, but any code can be used without departing from the scope of this disclosure.

[0237]

[0273] Figure 23B shows an example of a set of GHZ resource states arranged such that the qubits to be transmitted to a given fusion gate are schematicly adjacent to one another, i.e., they are pre-routed. In this figure, for adjacent qubits, each fusion can be performed between the pair of qubits (also referred to herein as the respective quantum subsystem, where each qubit from the pair of qubits input to the fusion site belongs to a different respective resource state). For example, at site 2302, two type II fusion measurements can be applied, one between qubits 2322 and 2324, and the other between qubits 2326 and 2328. It should be noted that before the fusion is performed, qubits 2322 and 2324 (or qubits 2326 and 2328) are not entangled with each other, but rather each is part of a separate resource state. Therefore, before the fusion measurement is performed, there is no large entangled cluster state known as a Lausendorf lattice.

[0238]

[0274] Referring to Figure 24A, a portion of the second layer of the underlying coding structure is shown as layer 2410 (corresponding to layer 2 shown in Figure 22C). In the FBQC system, to process a single layer as shown in Figure 24B, the FBQC method proceeds along the same lines described above with reference to Figures 23A and 23B, so the details will not be repeated here.

[0239]

[0275] Figures 25A to 25E illustrate in further detail a method for performing FBQC according to one or more embodiments. More specifically, the method described herein includes a step for performing a batch measurement of a particular quantum error correction code according to several embodiments, wherein different layers of the code are generated at different time steps (clock cycles) as described above with reference to Figures 23 to 24 and may be entangled together to provide a fused measurement to extract the syndrome information necessary to perform quantum error correction. As with other examples provided herein, a Raussendorf lattice is used for illustrative purposes, but other codes may be used without departing from the scope of this disclosure.

[0240]

[0276] For example, Figures 25A and 25B show portions of layers 1 and 3, and layers 2 and 4, respectively, from the Raussendorf lattice (referred to here as a quantum error correction (QEC) code) in Figure 22C. Figures 25C and 25D illustrate methods for processing these layers in an FBQC system and include exemplary resource states that can be used. As an example, the explanation is limited to vertices 1, 2, 3, and 4 of the QEC code, and the example focuses on how resource state generation and measurement are performed in an FBQC system.

[0241]

[0277] Returning to Figure 25A, in step 2501, a first set of resource states is provided during the first clock cycle. Figure 25D shows an example where, instead of single qubits being provided to vertices 1, 2, 3, 4, 5, etc., with those single qubits entangled across the lattice (as in the MBQC system), two or more qubits are provided, each originating from different, unentangled resource states, e.g., resource states A, B, C, D, E, F, and G. As used herein, the notation Aij is used to indicate the jth qubit from the a-th resource state of the i-th layer. For example, the a-th resource state of layer 1 in Figure 25D is a GHZ state containing four qubits labeled A11, A12, A13, and A14, as shown. Similarly, qubits containing resource state B provided as part of layer 1 could be labeled B11, B12, B13, and B14 (although labels are not explicitly shown in the figure at this point to avoid cluttering the figure). The qubits that are fused to generate syndrome information associated with vertices 1, 2, 3, 4, and 5 are also shown in Figure 25D, enclosed by solid ellipses 1, 2, 3, and 4. As used herein, each of these vertices is associated with hardware for performing type II fusion at the fusion site, as previously described.

[0242]

[0278] In some embodiments, resource states of any given layer can be generated / provided by a qubit entanglement system as described above with reference to Figure 3. However, as will be apparent to those skilled in the art who are interested in this disclosure, any qubit entanglement system can be used, and a given qubit entanglement system can use many different types of resource state generators and even generate different types of resource states. In this sense, the FBQC system is not entirely dependent on the selection of resource states and architecture for the qubit entanglement system, or even the architecture of the qubits themselves, thereby leaving system designers with great flexibility to implement a system that provides the best threshold against common error / noise sources.

[0243]

[0279] In step 2503, fusion instructions in the form of classical data (also referred to herein as fusion patterns) are provided to the fusion site. Referring again to Figure 3, for example, fusion pattern data frame 317 is an example of a set of fusion instructions (e.g., Type II fusion measurements on an XX basis) that can be applied between pairs of qubits from different entangled resource states in the fusion site during a particular clock cycle when a quantum application is executed on an FBQC system. As previously mentioned, in some embodiments, several fusion pattern data frames can be stored in memory as classical data. In some embodiments, a fusion pattern data frame can indicate whether an XX Type II fusion is applied to a particular fusion gate in the fusion site (or whether any other type of fusion is applied). Furthermore, a fusion pattern data frame can indicate that a Type II fusion is performed on different basis bases, e.g., XX, XY, ZZ, etc.

[0244]

[0280] Returning to Figure 25D, the fusion instruction for layer 1 can include fusion parameters (qubit positions and bases) for fusing two or more qubits from different resource states (referred to herein as each quantum subsystem, since each qubit exists in or is part of a separate resource state). For example, for fusion site 1, the fusion instruction can specify fusion parameters indicating that an XX type II fusion is performed between qubits from resource states A1, B1, and C1 (similarly for site 3 between E1, F1, and G1). More specifically, the two type II fusions performed at fusion site 1 can be specified as being between A14 and B12 and between C11 and B13. Similar instructions are provided for other fusion sites within the layer. For example, for fusion site 2, the fusion instruction can specify fusion parameters indicating that an XX type II fusion is performed between qubits from resource states B1, D1, and F1. More specifically, the two type II fusions performed at fusion site 2 can be specified as being between B14 and D12 and between D13 and F14. However, unlike fusion site 1 where all qubits are measured, fusion site 2 contains qubits that remain unmeasured until the second clock cycle. This is because the underlying structure of the QEC lattice requires that the quantum state of this qubit be preserved until it is fused with qubits from different layers in different clock cycles; that is, if this is the MBQC scheme, the qubit associated with this vertex will be entangled with qubits from other layers, for example, qubits 2 and 6 shown in Figures 25B and 25C, respectively.

[0245]

[0281] Returning to the explicit example shown in Figure 25D, the fusion instruction can specify that D14 will not be measured until the next clock cycle, where D14 is fused from a later layer, e.g., a layer 2 qubit shown in Figure 25E. In a photonic embodiment, the optical fiber can implement qubit delays for the above function, serving as a reliable quantum memory for storing qubits until needed in a future clock cycle. As used herein, these unmeasured (delayed) qubits are referred to as unmeasured quantum subsystems.

[0246]

[0282] Moving on to fusion section 4, this section is an example that includes interlayer fusion, i.e., fusion of a qubit from a resource state generated in this clock cycle with a qubit from a resource state generated in the previous clock cycle but not measured at that time, but instead delayed until the next clock cycle, or equivalently stored. In the case of fusion section 4, the fusion instruction can specify fusion parameters indicating that an XX type II fusion is performed between qubits from resource states of three different layers C1, B0, and B2. The fusion instruction can also include instructions to delay (not measure) qubits C12 and C13 until the next clock cycle. For example, in this case, the fusion instruction could indicate that in the next time step, C12 should be fused with B04 and C13 should be fused with B21.

[0247]

[0283] In step 2503, the fusion operation specified by the fusion instruction is performed, thereby generating classical data in the form of fusion measurement results. As previously described with reference to Figure 3 and equation (2), this classical data is then passed to the decoder and used to construct a syndrome graph to be used for quantum error correction.

[0248]

[0284] These examples are illustrative. The selection of error correction codes determines the set of qubit pairs fused from a particular resource state, such that the output of the qubit fusion system is classical data that can directly construct a syndrome graph. In some embodiments, classical error syndrome data is generated directly from the qubit fusion system without the need to perform additional single-particle measurements on any remaining qubits. In some embodiments, a batch measurement performed in the qubit fusion system destroys the qubit on which the batch measurement is performed.

[0249]

[0285] Introduction of qubits and path coding

[0250]

[0286] The dynamics of quantum objects, such as photons, electrons, atoms, ions, molecules, and nanostructures, follow the rules of quantum theory. More specifically, in quantum theory, the quantum state of a quantum object, such as a photon, is described by a set of physical properties, the entire set of which is called a mode. In some embodiments, a mode is defined by specifying the value (or distribution of values) of one or more properties of the quantum object. For example, in the case of a photon, as before, a mode can be defined by the photon's frequency, the photon's position in space (e.g., which waveguide or superposition of waveguides the photon is propagating through), the associated propagation direction (e.g., the photon's k-vector in free space), the photon's polarization state (e.g., the direction of the photon's electric and / or magnetic field (horizontal or vertical)), and so on.

[0251]

[0287] For photons propagating through waveguides, it is convenient to represent the state of a photon as one of a set of discrete spatiotemporal modes. For example, the spatial mode ki of a photon is determined according to one of a finite set of discrete waveguides through which the photon can propagate. Furthermore, the time mode tj is determined by which of a set of discrete time periods (referred to herein as "bins") the photon may exist. In some embodiments, the time discretization of the system can be provided by the timing of a pulsed laser involved in the generation of the photon. In the following examples, spatial modes are used primarily to avoid complicating the explanation. However, as those skilled in the art will see, the system and method can be applied to any type of mode, e.g., time modes, polarization modes, and any other modes or sets of modes that help define the quantum state. Furthermore, the following description describes embodiments that use a photonic waveguide to define the spatial mode of a photon. However, as those skilled in the art who are interested in this disclosure will see, any type of mode, e.g., polarization modes, time modes, etc., can be used without departing from the scope of this disclosure.

[0252]

[0288] For quantum systems of multiple indistinguishable particles, it is useful to describe the quantum state of the entire many-body system using the form of Fock states (sometimes called occupation representation), rather than describing the quantum state of each particle in the system. In the description of Fock states, a many-body quantum state is determined by how many particles there are in each mode of the system. Since a mode is a complete set of properties, this description is sufficient. For example, the multimode two-particle Fock state |1001> 1,2,3,4specifies a two-particle quantum state having one photon in mode 1, zero photons in mode 2, zero photons in mode 3, and one photon in mode 4. Again, as noted above, a mode can be any set of properties of a quantum object (and can depend on the single-particle basis states used to define the quantum state). In the case of photons, any two modes of the electromagnetic field can be used, and for example, the system can be designed to use modes associated with degrees of freedom that can operate passively in a linear optical system. For example, polarization, spatial degrees of freedom, or angular momentum can be used. For example, the two-particle Fock state |1001〉 1,2,3,4 The four-mode system represented by 1,2,3,4 can be physically implemented as four separate waveguides, with one photon moving inside two of the four waveguides (representing mode 1 and mode 4 respectively). Other examples of states of such a many-body quantum system are the four-photon Fock state |1111〉 1,2,3,4 representing each waveguide containing one photon, and the four-photon Fock state |2200〉 1,2,3,4 representing waveguide 1 and 2 each containing two photons, and waveguide 3 and 4 containing zero photons. For modes in which there are no photons, the term "vacuum mode" is used. For example, in the case of the four-photon Fock state |2200〉 1,2,3,4 modes 3 and 4 are referred to herein as "vacuum modes" (also called "ancilla modes").

[0253]

[0289] As used herein, a "qubit" (or quantum bit) is a physical quantum system with associated quantum states that can be used to encode information. A qubit, in contrast to a classical bit, can have a state that is a superposition of logical values such as 0 and 1. In some embodiments, the qubit is "dual-rail encoded" such that the logical value of the qubit is encoded by the occupancy of one of two modes by exactly one photon (a single photon). For example, consider two spatial modes of a photonic system associated with two separate waveguides. In some embodiments, the values of logical 0 and 1 can be encoded as follows. |0〉 L =|10〉 1,2 (3) |1〉 L =|01〉 1,2 (4)

[0254]

[0290] Here, the subscript "L" indicates that the ket represents a logical value (e.g., a qubit value), and the notation |ij〉 on the right side of the above equations (3) to (4) 1,2 indicates that there are i photons in the first waveguide and j photons in the second waveguide, respectively (e.g., i and j are integers). In this notation, the logical value |01〉 L (representing the state of two qubits, where the first qubit is in the "0" logical state and the second qubit is in the "1" logical state) of the two - qubit state can be represented using photon occupancy across four distinct waveguides as |1001〉 1,2,3,4 (i.e., 1 photon in the first waveguide, 0 photons in the second waveguide, 0 photons in the third waveguide, and 1 photon in the fourth waveguide). Throughout this disclosure, in some examples, various subscripts are omitted to avoid unnecessary mathematical clutter.

[0255]

[0291] XIV. Introduction to LOQC A. Dual - rail Photonic Qubits Qubits (and operations on qubits) can be implemented using a variety of physical systems. In some examples described herein, qubits are provided in an integrated photonic system using waveguides, beam splitters (or directional couplers), photonic switches, and single-photon detectors, and the modes that can be occupied by photons are spatiotemporal modes corresponding to the presence of photons in the waveguide. Modes can be coupled using mode couplers, such as optical beam splitters, to perform conversion operations, and measurement operations can be performed by coupling a single-photon detector to a particular waveguide. As those skilled in the art will see when accessing this disclosure, modes defined by any suitable set of degrees of freedom, such as polarization modes, time modes, etc., can be used without departing from the scope of this disclosure. For example, in the case of modes that differ only in polarization (e.g., horizontal (H) and vertical (V)), the mode coupler can be any optical element that coherently rotates the polarization, such as a birefringent material like a waveplate. In the case of other systems such as ion trap systems or neutral atom systems, the mode coupler can be any physical mechanism capable of coupling two modes, such as a pulsed electromagnetic field tuned to couple two internal states of an atom / ion.

[0256]

[0292] In some embodiments of photonic quantum computing systems using dual-rail coding, a pair of waveguides can be used to implement qubits. Figure 26A shows two representations (2600, 2600') of a pair of waveguides 2602, 2604 that may be used to give a dual-rail coded photonic qubit. In 2600, photon 2606 is in waveguide 2602 and the photon is not in waveguide 2604 (also called the vacuum mode), which in some embodiments corresponds to the |0> state of the photonic qubit. In 2600', photon 2608 is in waveguide 2604 and the photon is not in waveguide 2602, which in some embodiments corresponds to the |1> state of the photonic qubit. To preprocess photonic qubits of known states, a photon source (not shown) can be coupled to one end of one of the waveguides. A photon source can operate to emit a single photon into the waveguide into which it is coupled, thereby pre-processing a photonic qubit of a known state. The photon travels through the waveguide, and by periodically operating the photon source, a quantum system can be formed in the same waveguide pair, having qubits whose logical states are mapped to different time modes of the photonic system. Furthermore, by providing multiple pairs of waveguides, a quantum system can be formed, having qubits whose logical states correspond to different spatiotemporal modes. It should be understood that the waveguides in such a system do not necessarily need to have a specific spatial relationship to one another. For example, these waveguides can be arranged in parallel, but they do not necessarily have to be parallel.

[0257]

[0293] Occupied modes can be formed by using a photon source to generate photons that propagate within a desired waveguide. The photon source can be a resonator-based photon source that emits photon pairs, also known as a herald single-photon source. In one example of such a photon source, the source is driven by a pump, such as an optical pulse, coupled to a system of optical resonators capable of generating photon pairs by a nonlinear optical process (e.g., spontaneous four-wave mixing (SFWM), spontaneous parametric down-conversion (SPDC), second-harmonic generation, etc.). Many different types of photon sources can be used. An example of a photon pair source is a microring-based spontaneous four-wave mixing (SPFW) herald photon source (HPS). However, the exact type of photon source used is not critical; any type of photon source using any process, such as SPFW, SPDC, or any other process, can be used. Other classes of photon sources that do not necessarily require nonlinear materials, such as quantum dot sources or those using atomic and / or artificial atomic systems such as color centers in crystals, can also be used. In some cases, the photon source may or may not be coupled to a photonic cavity, for example, in the case of artificial atomic systems such as quantum dots coupled to a cavity. Other types of photon sources, such as photomechanical systems, also exist in SPWM and SPDC.

[0258]

[0294] In such cases, the operation of the photon source can be deterministic or nondeterministic (sometimes called “probabilistic”), such that a given pump pulse may or may not generate a pair of photons. In some embodiments, coherent spatial and / or temporal multiplexing (referred to herein as “active” multiplexing) of several nondeterministic photon sources can be used so that the probability of one mode being occupied during a given cycle approaches 1. As will be apparent to those skilled in the art, many different active multiplexing architectures incorporating spatial and / or temporal multiplexing can be envisioned. For example, active multiplexing schemes can be used that employ logarithmic trees, generalized Mach-Zehnder interferometers, multimode interferometers, chain sources, dump-to-pump chain sources, asymmetric polycrystalline single-photon sources, or any other type of active multiplexing architecture. In some embodiments, the photon source can employ active multiplexing schemes, such as those employing quantum feedback control.

[0259]

[0295] The measurement operation can be performed by coupling a waveguide to a single-photon detector that generates a classical signal (e.g., a digital logic signal) indicating that a photon has been detected by the detector. Any type of photodetector with sensitivity to single photons can be used. In some embodiments, the detection of a photon (e.g., at the output end of the waveguide) may indicate an occupied mode, while the absence of a detected photon may indicate an unoccupied mode. In some embodiments, the measurement operation is performed on a specific basis (e.g., a basis defined by one of the Pauli matrices, called X, Y, or Z), and the qubit can be transformed to a specific basis by applying mode coupling as described later.

[0260]

[0296] The following embodiments relate to physical embodiments of unitary transformation operations that combine the modes of a quantum system, which can be understood as transforming the quantum state of the system. For example, if the initial state of a quantum system (before mode combination) is one in which one mode is occupied with probability 1 and another in which one mode is not occupied with probability 1 (e.g., a state |10〉 in Fock notation, where numbers indicate the occupation of each state), then mode combination can result in a state with a non-zero probability in which both modes are occupied, e.g., state a1|10〉+a2|01〉, where |a1| 2 +|a2| 2 = 1. In some embodiments, this type of operation can be achieved by using a beam splitter to couple modes together and a variable phase shifter to apply a phase shift to one or more modes. The amplitudes a1 and a2 depend on the reflectance (or transmittance) of the beam splitter and any phase shift introduced.

[0261]

[0297] Figure 26B shows a schematic diagram 2610 (also called a circuit diagram or circuit notation) for coupling two modes. The modes are depicted as horizontal lines 2612 and 2614, and the mode coupler 2616 is indicated by vertical lines terminated with nodes (solid dots) to identify the coupled modes. In the more specific language of linear quantum optics, the mode coupler 2616 shown in Figure 26B represents a 50 / 50 beam splitter that implements the transfer matrix.

number

number

number

[0262]

[0298] For example, the application of the mode coupler shown in Figure 26B results in the following mapping.

number

[0263]

[0299] Therefore, the action of the mode coupler described by equation (4) is to take the input states |10〉, |01〉 and |11〉.

number

[0264]

[0300] Figure 26C shows a physical implementation of mode coupling that implements the transfer matrix T of equation (4) with respect to two photonic modes according to several embodiments. In this example, mode coupling is implemented using a waveguide beam splitter 2620, sometimes called a directional coupler or mode coupler. The waveguide beam splitter 2620 can be realized by placing two waveguides 2622, 2624 close enough so that the evanescent field of one waveguide can be coupled to the other waveguide. Different couplings between modes can be obtained by adjusting the spacing d between waveguides 2622, 2624 and / or the length l of the coupling region. In this way, the waveguide beam splitter 2620 can be configured to have a desired transmittance. For example, the beam splitter can be designed to have a transmittance equal to 0.5 (i.e., a 50 / 50 beam splitter to implement a particular form of the transfer matrix T introduced above). If other transfer matrices are desired, the reflectance (or transmittance) can be designed to be greater than 0.6, greater than 0.7, greater than 0.8, or greater than 0.9 without departing from the scope of this disclosure.

[0265]

[0301] In addition to mode coupling, some unitary transformations may include phase shifts applied to one or more modes. In some photonic implementations, a variable phase shifter can be implemented in an integrated circuit to control the relative phase of the photon states diffused across multiple modes. An example of a transfer matrix defining such a phase shift is given below (for applying +i and -i phase shifts to the second mode, respectively).

number

[0266]

[0302] In the case of silica-on-silicon materials, several embodiments implement a variable phase shifter using a thermo-optic switch. The thermo-optic switch changes the waveguide temperature by 10°C via the thermo-optic effect. -5A resistive element fabricated on the surface of a chip is used, which can be used to change the refractive index n by increasing it by about K. As those skilled in the art who have access to this disclosure will see, a variable, electrically tunable phase shift can be generated using any effect that changes the refractive index of a portion of the waveguide. For example, some embodiments use beam splitters based on any material that supports the electro-optic effect, so-called χ2 and χ3 materials, such as lithium niobate, BBO, KTP, BTO, PZT, etc., or even doped semiconductors, such as silicon, germanium, etc.

[0267]

[0303] B. Photonic Mode Coupler: Beam Splitter Beam splitters with variable transmittance and arbitrary phase relationships between output modes can also be achieved by combining a directional coupler and a variable phase shifter in a Mach-Zehnder interferometer (MZI) configuration 2630, for example, as shown in Figure 26D. Complete control over the relative phase and amplitude of the two modes 2632a and 2632b in dual-rail coding can be achieved by varying the phases provided by phase shifters 2636a, 2636b, and 2636c, as well as the length and proximity of the coupling regions 2634a and 2634b. Figure 26E shows a slightly simpler example of an MZI 2640 that enables variable transmittance between modes 2632a and 2632b by varying the phase provided by phase shifter 2637. Figures 26D and 26E are examples of ways in which mode couplers can be implemented in physical devices, but any type of mode coupler / beam splitter can be used without departing from the scope of this disclosure.

[0268]

[0304] In some embodiments, a beam splitter and a phase shifter can be used in combination to implement various transfer matrices. For example, Figure 27A shows a mode coupler 2700 implementing the following transfer matrices in a schematic form similar to that of Figure 26A.

number

[0269]

[0305] Therefore, the mode coupler 2700 applies the following mapping.

number

[0270]

[0306] Transfer matrix T in equation (10) r This relates to the transfer matrix T in equation (4) due to the phase shift in the second mode. This is schematically shown in Figure 27A by the closed node 2707 (line 2712) where the mode coupler 2716 is coupled to the first mode and the open node 2708 (line 2714) where the mode coupler 2716 is coupled to the second mode. More specifically, T r = sTs, and as shown on the right side of Figure 27A, the mode coupler 2716 can be implemented using the mode coupler 2716 (as described above) with leading and trailing phase shifts (shown by white squares 2718a and 2718b). Therefore, the transfer matrix T r This can be implemented by a physical beam splitter as shown in Figure 27B, in which case the white triangle represents the +i phase shifter.

[0271]

[0307] C. Example of a photonic diffusion circuit A network of mode couplers and phase shifters can be used to perform coupling between three or more modes. For example, Figure 28 shows a four-mode coupling scheme that performs a “spreader” or “mode information erasure” transform on four modes, i.e., this scheme delocalizes the photon among each of the four output modes such that the probability of the photon being captured in any one of the input modes and detected in any one of the four output modes is equal. (The well-known Hadamard transform is an example of a spreader transform.) As shown in Figure 26A, horizontal lines 2812–2815 correspond to modes, and mode coupling is indicated by vertical lines 2816 with nodes (dots) to identify the coupled modes. In this case, four modes are coupled. Circuit notation 2802 is an equivalent representation of schematic 2804, which is a network of first-order mode couplings. More generally, if higher-order mode coupling can be performed as a network of first-order mode couplings, a circuit notation similar to notation 2802 (with an appropriate number of modes) may be used.

[0272]

[0308] Figure 29 shows an example of an optical device 2900 capable of performing four-mode spread conversion schematically shown in Figure 28 according to several embodiments. The optical device 2900 includes a first set of optical waveguides 2901, 2903 formed in a first material layer (represented by solid lines in Figure 29) and a second set of optical waveguides 2905, 2907 formed in a separate second material layer different from the first material layer (represented by dashed lines in Figure 29). The second and first material layers are located at different heights on the substrate. As will be apparent to those skilled in the art, if appropriate low-loss waveguide crossings are used, an interferometer like the one shown in Figure 29 can be implemented in a single layer.

[0273]

[0309] At least one optical waveguide 2901,2903 of the first set of optical waveguides is coupled with optical waveguides 2905,2907 of the second set of optical waveguides using a suitable optical coupler of any kind. For example, the optical device shown in Figure 29 includes four optical couplers 2918,2920,2922,2924. Each optical coupler may have a coupling region in which the two waveguides propagate in parallel. The two waveguides are shown in Figure 29 as being offset from each other within the coupling region, but the two waveguides may be positioned directly above and directly below each other within the coupling region without any offset. In some embodiments, one or more of the optical couplers 2918, 2920, 2922, and 2924 are configured to have a coupling efficiency of approximately 50% between two waveguides (e.g., 49%–51%, 49.9%–50.1%, 49.99%–50.01%, and 50%). For example, the lengths of the two waveguides, the refractive indices of the two waveguides, the widths and heights of the two waveguides, the refractive index of the material located between the two waveguides, and the distance between the two waveguides are selected to give a 50% coupling efficiency between the two waveguides. This allows the optical coupler to operate like a 50 / 50 beam splitter.

[0274]

[0310] Furthermore, the optical device shown in Figure 29 may include two interlayer optical couplers 2914 and 2916. Optical coupler 2914 enables the transmission of light propagating through a waveguide on a first material layer to a waveguide on a second material layer, and optical coupler 2916 enables the transmission of light propagating through a waveguide on the second material layer to a waveguide on the first material layer. Optical couplers 2914 and 2916 enable the use of optical waveguides located in at least two different layers in a multi-channel optical coupler, thereby enabling a compact multi-channel optical coupler.

[0275]

[0311] Furthermore, the optical device shown in Figure 29 includes an uncoupled waveguide intersection region 2926. In some implementations, two waveguides (2903 and 2905 in this example) intersect each other without a parallel coupling region at the intersection within the uncoupled waveguide intersection region 2926 (for example, the waveguides may be two straight waveguides intersecting each other at an angle of approximately 90 degrees).

[0276]

[0312] As those skilled in the art will see, the examples described above are illustrative, and many different transformation matrices can be implemented using photonic circuits employing beam splitters and / or phase shifters, including transformation matrices for real and imaginary Hadamard transforms of any order, discrete Fourier transforms, and so on. One class of photonic circuits, referred to herein as “spreaders” or “mode information erasure (MIE)” circuits, has the property that, given an input is a single photon localized to one input mode, the circuit delocalizes a photon among several output modes such that the photon has an equal probability of being detected in any one of the output modes. Examples of spreader or MIE circuits include circuits that implement Hadamard transfer matrices. (It should be understood that a spreader or MIE circuit may receive inputs that are not single photons localized in one input mode, and the behavior of the circuit in such cases depends on the specific transporter matrix implemented.) In other examples, a photonic circuit may implement other transporter matrices that, with respect to a single photon in one input mode, have different probability matrices of detecting the photon in different output modes that are not equal.

[0277]

[0313] D. Exemplary photonic battery state generation circuit A Bell pair is a pair of qubits in any type of maximally entangled state called a Bell state. For dual-rail coded qubits, examples of Bell states (also called Bell ground states) include:

number

[0278]

[0314] In a computational basis with two states (e.g., a logical basis), the Greenberger-Horne-Zeilinger state is the quantum superposition of all the qubits in the first state of the two states superimposed with all the qubits in the second state. Using the aforementioned logical basis, a general M-qubit GHZ state can be written as follows:

number

[0279]

[0315] In some embodiments, entangled states of multiple photonic qubits can be formed by combining modes of two (or more) qubits and performing measurements in other modes. As an example, Figure 30 shows a schematic of a Bell state generator 3000 that can be used in several dual-rail coding photonic embodiments. In this example, modes 3032(1)-3032(4) are initially occupied by photons (indicated by dashed lines), while modes 3032(5)-3032(8) are initially vacuum modes. (As those skilled in the art will see, other combinations of occupied and unoccupied modes can be used.)

[0280]

[0316] A first-order mode coupling (e.g., implementing the transfer matrix T of equation (4)) is performed on pairs of occupied and unoccupied modes, as shown by mode couplers 3031(1)-3031(4). Subsequently, a mode information elimination coupling (e.g., performing a four-mode spread-mode transform as shown in Figure 13) is performed on four of the modes (modes 3032(5)-3032(8)), as shown by mode coupler 3037. Modes 3032(5)-3032(8) act as "herald" modes, which are measured and used to determine whether the Bell state was successfully generated in the other four modes 3032(1)-3032(4). For example, detectors 3038(1)-3038(4) can be coupled to modes 3032(5)-3032(8) after the second-order mode coupler 3037. Each detector 3038(1)-3038(4) can output a classical data signal (e.g., a voltage level on a conductor) indicating whether it has detected a photon (or the number of photons detected). These outputs can be coupled to a classical decision logic circuit 3040 that determines, based on the classical output data, whether a Bell state exists in the other four modes 3032(1)-3032(4). For example, the decision logic circuit 3040 can be configured such that a Bell state is confirmed (also called a "success" for the Bell state generator) only if a single photon is detected by exactly two of the detectors 3038(1)-3038(4). Modes 3032(1)-3032(4) can be mapped to logic states of two qubits (qubit 1 and qubit 2), as shown in Figure 30. Specifically, in this example, the logical state of qubit 1 is based on the occupancy rates of modes 3032(1) and 3032(2), and the logical state of qubit 2 is based on the occupancy rates of modes 3032(3) and 3032(4). It should be noted that the operation of the Bell state generator 3000 can be nondeterministic. That is, inputting four photons as shown in the figure does not guarantee that a Bell state will be generated in modes 3032(1)-3032(4). In one embodiment, the probability of success is 4 / 32.

[0281]

[0317] In some embodiments, it is desirable to form resource states of multiple entangled qubits (typically three or more qubits, although a Bell state can be understood as a resource state of two qubits). One technique for forming larger entangled systems is by using a "fusion" gate. A fusion gate takes two input qubits, each of which is typically part of an entangled system. The fusion gate performs a "fusion" operation on the input qubits, producing either one ("Type I fusion") or zero ("Type II fusion") output qubits, such that the first two entangled systems are fused into a single entangled system. Fusion gates can be used to generate entanglement between qubits and are concrete examples of a common class of two-particle projection measurements that are particularly well-suited to photonic architectures. Examples of Type I and Type II fusion gates are described below.

[0282]

[0318] E. Example of a fusion gate optical circuit Figures 31–36 show several embodiments of photonic circuit implementations of fusion gates or fusion circuits for photonic qubits that can be used according to several embodiments using Type II fusion. It should be understood that these exemplary embodiments are illustrative and not limiting. More generally, as used herein, the term “fusion gate” refers to a device capable of performing two-particle projection measurements, e.g., Bell projections that can measure two operators, e.g., operator XX, ZZ, operator XX, ZY, etc., depending on a selected Bell basis. In polarization coding, a Type II fusion circuit (or gate) takes two input modes, mixes them with a polarizing beam splitter (PBS), then rotates each of them by 45 degrees, and then measures them with a computational basis. An example is shown in Figure 31. In path coding, a Type II fusion circuit takes four modes, swaps the second and fourth, applies a 50:50 beam splitter between two pairs of adjacent modes, and then detects them all. An example is shown in Figure 32.

[0283]

[0319] Fusion gates can be used to construct larger entangled states by utilizing the so-called "redundant coding" of qubits. This resides in a single qubit represented by multiple photons. That is,

number

[0284]

[0320] Therefore, a logical qubit is encoded by n individual qubits. This is achieved by measuring adjacent qubits in an X basis.

[0285]

[0321] This encoding, represented graphically as n qubits with no edges in between (as shown in Figure (b) of Figure 33), has the advantage that Pauli measurements for redundant qubits do not split the cluster, but rather remove the measured photons from the redundant encoding, combine the neighboring qubits into a single qubit that inherits the combination of the input qubits, and possibly add a phase. Furthermore, another advantage of this type of fusion is that it is loss-tolerant. Since both modes are measured, there is no way to obtain a detection pattern that signals success if one of the photons is lost. Finally, type II fusion does not require distinction between different photon counts, as two detectors only need to click to signal a successful fusion, which can only occur if the photon count in each detector is 1.

[0286]

[0322] In polarization coding, if a single photon is detected at each detector, fusion has a 50% probability of success. In this case, polarization coding effectively performs Bell state measurements on the qubits transmitted through it, projecting pairs of logical qubits into the most entangled state. If a gate fails (as indicated by 0 or 2 photons in one of the detectors), a measurement is performed on each photon in the computational basis, removing them from redundant coding but without destroying the logical qubits. The effect of fusion on cluster generation is shown in Figure 33, where (a) shows the measurement of qubits in a linear cluster in the X basis to couple qubits with their neighbors to form a single logical qubit, and (c) and (c') show the effect of gate success and failure on the cluster structure. It can be seen that a 2D cluster can be constructed when fusion is successful.

[0287]

[0323] The correspondence between the detection pattern and the Kraus operator performed by the gate on the state can be searched. In this case, both qubits are detected, so these are projectors.

number

[0288]

[0324] The first two rows correspond to the "success" outcome, projecting the two qubits into a bell state, while the bottom two rows project to the "failure" outcome, in which case the two qubits are projected into a product state.

[0289]

[0325] In some embodiments, the success rate of type II fusion can be increased by using auxiliary Bell pairs or pairs of single photons. Using a single ancillary pair or two pairs of single photons can boost the success rate to 75%.

[0290]

[0326] One technique used to boost a fusion gate derives from the recognition that, when successful, it is equivalent to a Bell state measurement on an input qubit. Therefore, increasing the success rate of a fusion gate corresponds to increasing the success rate of the Bell state measurement it performs. Two different techniques for improving the probability of identifying Bell states have been developed by Grice (using Bell pairs) and Ewert & van Loock (https: / / arxiv.org / pdf / 1403.4841.pdf) (using single photons).

[0291]

[0327] The former demonstrated that the auxiliary bell pair could achieve a 75% success rate, and theoretically, the procedure could be repeated using increasingly complex interferometers and larger entangled states to reach any given success rate. However, the complexity of the circuit and the size of the required entangled states could make this impractical.

[0292]

[0328] The second technique boosts the success rate to 75% by utilizing four single photons input in two modes with opposite polarizations. It has also been numerically shown that repeating this procedure twice can yield a 78.125% success rate, but it has not been shown that any other method can arbitrarily increase the success rate.

[0293]

[0329] Figure 34 shows a Type II fusion gate boosted once using these two techniques for both polarization and path coding. The success probability for both circuits is 75%.

[0294]

[0330] The detection patterns that indicate successful fusion are described below for two types of circuits.

[0295]

[0331] When Bell states are used to boost fusion, the logic behind the “success” detection pattern is best understood by considering two pairs: a group corresponding to the input photon modes (polarization modes 1 and 2, and the top four modes of path coding) and a group corresponding to the Bell pair input modes (polarization modes 3 and 4, and the bottom four modes of path coding). These are referred to as the “main” and “ancilla” pairs, respectively. Then, (a) whenever a total of four photons are detected, fusion success is indicated, and (b) fewer than four photons are detected in each group of the detector.

[0296]

[0332] When four single photons are used as auxiliary resources, the gate success is indicated by (a) when a total of six photons are detected, and (b) when fewer than four photons are detected in each detector.

[0297]

[0333] If the gate is successful, the two input qubits are projected onto one of the four bell pairs, because they can all be distinguished from one another by the use of auxiliary resources. The specific projection depends on the resulting detection pattern, as mentioned earlier.

[0298]

[0334] Both boosted Type II fusion circuits, designed to take one Bell pair and four single photons as ancillas respectively, can be used to perform Type II fusion with a variable success probability when no ancillas are present, or when only some of them are present (in the case of four single-photon ancillas). This is particularly useful as it allows fusion to be performed in a flexible way using the same circuit depending on the available resources. If ancillas are present, they can be input into the gate to boost the probability of successful fusion. However, if they are not present, the gate can be used to attempt fusion with a lower but non-zero success probability.

[0299]

[0335] As far as fusion gates boosted using a single Bell pair are concerned, the only case to consider is the absence of an ancilla. In this case, the logic of the detection pattern that signals success can be understood by again considering the detectors of the aforementioned pair: (a) if two photons are detected by different detectors, the fusion is still successful, and (b) if one photon is detected by the "main" pair detector and one photon by the "ancilla" pair detector.

[0300]

[0336] In the case of a circuit boosted using four single photons, multiple modifications are possible, and all or part of the ancilla can be removed. This is similar to a boosted bell state generator based on the same principle.

[0301]

[0337] First, consider the case where no ancilla exists at all. As expected, fusion succeeds with a 50% probability, which is the success rate of unboosted fusion. In this case, fusion succeeds whenever two photons are detected by any two separate detectors.

[0302]

[0338] Regarding boosted BSGs, the presence of an odd number of ancilla was found to be detrimental to the gate's success rate; with one photon present, the gate only succeeds 32.5% of the time, while with three photons present, the success rate is 50%, the same as when it is not boosted.

[0303]

[0339] If only two of the four Ancilla exist, two effects are possible.

[0304]

[0340] When these are inputs in different modes in polarization coding, i.e., different adjacent pairs of ancillary modes in path coding, the probability of success drops to 25%.

[0305]

[0341] However, if two ansciras are input in the same polarization mode, i.e., in the same adjacent mode pair in path coding, the probability of success is boosted to 62.5%. In this case, the pattern indicating success can again be understood by grouping the detectors into two pairs, i.e., a pair of branches in the circuit into which the ansciras are input (Group 1) and a pair of other branches (Group 2). This distinction is particularly clear in the polarization coding diagram. Considering these groups, if the fusion is successful, (a) four photons are detected overall; (b) fewer than four photons are detected in each detector of Group 1; (c) fewer than two photons are detected in each detector of Group 2.

[0306]

[0342] In these examples, the fusion gate works by projecting the input qubits into the most entangled state when successful. The basis into which such a state is encoded can be modified by introducing a local rotation of the input qubits before they enter the gate, i.e., before they are mixed in the PBS in polarization coding. Changing the polarization rotation of a photon before it interferes in the PBS yields a different subspace into which the photon state is projected, resulting in a different fusion behavior for the cluster state. In path coding, this corresponds to applying a local beam splitter or a combination of a beam splitter and phase shift corresponding to a desired rotation between pairs of modes constituting the qubit (adjacent pairs in the diagram above).

[0307]

[0343] This can be useful in implementing different types of cluster operations in both successful and unsuccessful scenarios, and it can be very useful in optimizing the construction of large cluster states from small entangled states.

[0308]

[0344] Figure 35 shows a table with the effects of several rotated deformations of a type II fusion gate used to fuse two small entangled states. The figure shows the gate in polarization coding, the effective projection performed, and the final effect on the cluster state.

[0309]

[0345] Rotations to different ground states are further shown in Figure 36, which illustrates an example of a photonic circuit for a Type II fusion gate implementation using path coding. Fusion gates for ZX fusion, XX fusion, ZZ fusion, and XZ fusion are shown. In each example, a combination of beam splitter and phase shifter (e.g., as shown above) can be used.

[0310]

[0346] As those skilled in the art will see, the embodiments described herein are illustrative and not limiting, and many modifications and variations are possible. The measurements performed and the conditions under which they operate can be selected such that the measurement results have redundancy that causes fault tolerance. For example, codes can be directly input with the measurements, or correlations can be generated with the measurements that directly address both the destructive nature and the entanglement-destructive nature of the measurements in a fault-tolerant manner. This can be treated as part of classical decoding, for example, a failed fusion operation can be treated as an erasure by a code.

[0311]

[0347] In relation to the attached diagram, components that may include memory may include persistent machine-readable media. As used herein, the terms “machine-readable media” and “computer-readable media” refer to any storage medium involved in providing data that causes a machine to operate in a particular manner. In the embodiments provided above, various machine-readable media may be involved in providing instructions / codes to a processor and / or other devices for execution. In addition to or instead of this, machine-readable media may be used to store and / or carry such instructions / codes. In many embodiments, computer-readable media are physical and / or tangible storage media. Such media can take many forms, including but not limited to non-volatile media, volatile media, and transmission media. Common forms of computer-readable media include, for example, magnetic and / or optical media, punch cards, paper tape, any other physical media having a pattern of holes, RAM, programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), FLASH-EPROM, any other memory chip or cartridge, carrier waves as described below, or any other media from which a computer can read instructions and / or codes.

[0312]

[0348] The methods, systems, and devices described herein are examples. Various embodiments may omit, substitute, or add various procedures or components as needed. For example, features described in relation to a particular embodiment can be combined in various other embodiments. Different aspects and elements of embodiments can be combined in similar ways. Various components of the figures provided herein can be implemented in hardware and / or software. Furthermore, technology is evolving, and therefore many elements are examples that do not limit the scope of this disclosure to those specific examples.

[0313]

[0349] Referring to signals as bits, information, values, elements, symbols, characters, variables, terms, numbers, digits, etc., has sometimes proven convenient, primarily for reasons of general use. However, all of these or similar terms should be associated with appropriate physical quantities and should be understood as merely convenient labels. Unless otherwise specified, as is evident from the above description, descriptions throughout this specification using terms such as “process,” “operate,” “calculate,” “determine,” “verify,” “identify,” “associate,” “measure,” and “execute” are understood to refer to the operation or process of a particular device, such as a dedicated computer or similar dedicated electronic computing device. Thus, in the context of this specification, a dedicated computer or similar dedicated electronic computing device can manipulate or convert signals that are typically expressed as physical electronic, electric, or magnetic quantities in the memory, registers, or other information storage devices, transmitting devices, or display devices of the dedicated computer or similar dedicated electronic computing device.

[0314]

[0350] As those skilled in the art will see, the information and signals used to communicate the messages described herein may be represented using any of a variety of different techniques and technologies. For example, the data, instructions, commands, information, signals, bits, symbols, and chips that may be referenced throughout the above description may be represented by voltage, current, electromagnetic waves, magnetic fields or particles, optical fields or particles, or any combination thereof.

[0315]

[0351] As used herein, the terms “and,” “or,” and “and / or” may have a variety of meanings, which are also expected to depend at least in part on the context in which such terms are used. In general, when “or” is used to relate a list such as A, B, or C, it is intended to mean A, B, and C as used here in an inclusive sense, as well as A, B, or C as used here in an exclusive sense. Furthermore, the term “one or more” as used herein may be used to describe any singular feature, structure, or property, or to describe several combinations of features, structures, or properties. However, it should be noted that this is merely illustrative, and the claimed subject matter is not limited to these examples. Furthermore, when the term “at least one of” is used to relate a list such as A, B, or C, it may be interpreted to mean any combination of A, B, and / or C, such as A, B, C, AB, AC, BC, AA, AAB, ABC, AABBCCC, etc.

[0316]

[0352] Throughout this specification, any reference to “one example,” “an example,” “a specific example,” or “a typical embodiment” means that a particular feature, structure, or characteristic described in relation to a feature and / or example may be included in at least one feature and / or example of the claimed subject matter. Therefore, the appearance of phrases such as “in one example,” “in one example,” “a specific example,” “in a particular embodiment,” or other similar phrases in various parts of this specification does not necessarily all refer to the same feature, example, and / or limitation. Furthermore, a particular feature, structure, or characteristic may be combined in one or more examples and / or features.

[0317]

[0353] In some implementations, operation or processing may involve the physical manipulation of physical quantities. Generally, but not always, such quantities may take the form of electrical or magnetic signals that can be stored, transferred, combined, compared, or otherwise manipulated. Referring to signals as bits, data, values, elements, symbols, characters, terms, digits, etc., has sometimes proven convenient, primarily for reasons of general use. However, all of these or similar terms should be associated with appropriate physical quantities and should be understood as merely convenient labels. Unless otherwise specified, as is evident from the description herein, descriptions using terms such as “processing,” “computing,” “calculating,” and “determining” throughout this specification are understood to refer to the operation or process of a particular device, such as a dedicated computer, dedicated computing device, or similar dedicated electronic computing device. Thus, in the context of this specification, a dedicated computer or similar dedicated electronic computing device can manipulate or convert signals that are typically represented as physical electronic or magnetic quantities in the memory, registers, or other information storage devices, transmitting devices, or display devices of the dedicated computer or similar dedicated electronic computing device.

[0318]

[0354] In the detailed description above, numerous specific details have been described to give a complete understanding of the subject matter described in the claims. However, as those skilled in the art will see, the subject matter described in the claims may be carried out without these specific details. In other examples, methods and apparatus known to those skilled in the art are not described in detail so as not to obscure the subject matter described in the claims. Therefore, it is intended that the subject matter described in the claims is not limited to the specific examples disclosed, but may also include all embodiments and equivalents thereof of such subject matter that fall within the scope of the appended claims.

[0319]

[0355] In embodiments including firmware and / or software, the methodology may be implemented in modules (e.g., procedures, functions, etc.) that perform the functions described herein. Machine-readable media that substantially embody instructions may be used when implementing the methodology described herein. For example, software codes may be stored in memory and executed by a processor unit. The memory may be implemented within the processor unit or outside the processor unit. As used herein, the term “memory” refers to long-term, short-term, volatile, non-volatile, or other types of storage media, and is not limited to any particular type of memory or the number of memories or the type of media in which the memory is stored.

[0320]

[0356] When implemented in firmware and / or software, the functionality may be stored as one or more instructions or codes on a computer-readable storage medium. Examples include computer-readable media encoded in data structures and computer-readable media encoded in computer programs. Computer-readable media include physical computer storage media. Storage media can be any available medium that can be accessed by a computer. Examples, but not limited to, such computer-readable media may include RAM, ROM, EEPROM, compact disk read-only memory (CD-ROM) or other optical disk storage devices, magnetic disk storage devices, semiconductor storage devices, or other storage devices, or any other medium that can be used to store desired program codes in the form of instructions or data structures and can be accessed by a computer. As used herein, disks and discs include compact disks (CDs), laser disks, optical disks, digital multipurpose disks (DVDs), floppy disks, and Blu-ray discs, where disks typically reproduce data magnetically and discs reproduce data optically using a laser. Combinations of the above should also be included within the scope of computer-readable media.

[0321]

[0357] In addition to storage on a computer-readable storage medium, instructions and / or data may be provided as signals on a transmission medium included in a communication device. For example, the communication device may include a transceiver having signals indicating instructions and data. The instructions and data are configured to cause one or more processors to perform the functions outlined in the claims. That is, the communication device includes a transmission medium having signals indicating information for performing the disclosed functions. At a first point in time, the transmission medium included in the communication device may include a first portion of the information for performing the disclosed functions, and at a second point in time, the transmission medium included in the communication device may include a second portion of the information for performing the disclosed functions.

Claims

1. A first input coupled to a first qubit and a first switch, wherein the first switch includes a first output, a second output, and a third output, A first single-qubit measurement device coupled to the first output of the first switch, A second single-qubit measurement device coupled to the first output of the second switch, A first two-qubit measurement device coupled to the second output of the first switch and the second output of the second switch, A second two-qubit measurement device coupled to the third output of the first switch and the third output of the second switch, A system equipped with these features.

2. The system according to claim 1, further comprising a fusion network controller circuit coupled to the first switch and the second switch.

3. The system according to claim 1, further comprising a decoder coupled to the output of the first single-qubit measurement device, the output of the second single-qubit measurement device, the output of the first two-qubit measurement device, and the output of the second two-qubit measurement device.

4. The system according to claim 1, wherein the first qubit is entangled with one or more other qubits as part of a first resource state, the second qubit is entangled with one or more other qubits as part of a second resource state, and none of the qubits from the first resource state are entangled with any of the qubits from the second resource state.

5. The system according to claim 1, wherein the first and second two-qubit measurement devices are configured to perform a destructive batch measurement on the first qubit and the second qubit and to output classical information representing the batch measurement result.

6. The system according to claim 1, wherein the first qubit and the second qubit are photonic qubits.

7. The system according to claim 6, wherein the coupling between the first and second qubits and the first and second switches includes a plurality of photonic waveguides.

8. The system according to claim 1, wherein the first single-qubit measurement device is configured to measure the first qubit in a Z basis.

9. The system according to claim 1, wherein the second single-qubit measurement device is configured to measure the second qubit in a Z basis.

10. The system according to claim 1, wherein the first two-qubit measurement device is configured to perform a projected Bell measurement between the first qubit and the second qubit.

11. The system according to claim 1, wherein the second two-qubit measurement device is configured to perform a projected Bell measurement between the first qubit and the second qubit.

12. The system according to claim 10, wherein the projection Bell measurement is a linear optical type II fusion measurement.

13. The system according to claim 11, wherein the projection Bell measurement is a linear optical type II fusion measurement.

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